{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0001","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4efe5735c26f00a91a693f39c488adee9fad8861eb116183056276bc88ca749b","family_id":"qf-mod47-mle-law-likelihood","seed":202611164001,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x0\",\"y0\"],\"observed\":\"x0\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a0\",\"b0\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a0, b0], observations=[x0, y0] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x0. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a0, b0], [x0, y0], [[3/4, 1/4], [1/4, 3/4]], x0, [[a0, 3/4], [b0, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a0, b0], [x0, y0], [[3/4, 1/4], [1/4, 3/4]], x0, [[a0, 3/4], [b0, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0002","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a84ba587c6b023c294c65cf08435cfbcb177a70d714c7fc8946053e5a362225","family_id":"qf-mod47-mle-law-likelihood","seed":202611164002,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x1\",\"y1\"],\"observed\":\"x1\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a1\",\"b1\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a1, b1], observations=[x1, y1] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x1. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a1, b1], [x1, y1], [[3/4, 1/4], [1/4, 3/4]], x1, [[a1, 3/4], [b1, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a1, b1], [x1, y1], [[3/4, 1/4], [1/4, 3/4]], x1, [[a1, 3/4], [b1, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0003","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6e8f5dddff9872b07388eb31967480d31125df4d0f691b000e0a9ff47f25ae9","family_id":"qf-mod47-mle-law-likelihood","seed":202611164003,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x2\",\"y2\"],\"observed\":\"x2\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a2\",\"b2\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a2, b2], observations=[x2, y2] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x2. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a2, b2], [x2, y2], [[3/4, 1/4], [1/4, 3/4]], x2, [[a2, 3/4], [b2, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a2, b2], [x2, y2], [[3/4, 1/4], [1/4, 3/4]], x2, [[a2, 3/4], [b2, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0004","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2598484bb7ffc33c31c6fb9997685645fa811ad3ff371b5434f4aa98da2e4e1c","family_id":"qf-mod47-mle-law-likelihood","seed":202611164004,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x3\",\"y3\"],\"observed\":\"x3\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a3\",\"b3\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a3, b3], observations=[x3, y3] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x3. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a3, b3], [x3, y3], [[3/4, 1/4], [1/4, 3/4]], x3, [[a3, 3/4], [b3, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a3, b3], [x3, y3], [[3/4, 1/4], [1/4, 3/4]], x3, [[a3, 3/4], [b3, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0005","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"290b5665b326e8a0e2856bcfc651030589ade5ec47510beade9335e841861628","family_id":"qf-mod47-mle-law-likelihood","seed":202611164005,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x4\",\"y4\"],\"observed\":\"x4\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a4\",\"b4\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a4, b4], observations=[x4, y4] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x4. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a4, b4], [x4, y4], [[3/4, 1/4], [1/4, 3/4]], x4, [[a4, 3/4], [b4, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a4, b4], [x4, y4], [[3/4, 1/4], [1/4, 3/4]], x4, [[a4, 3/4], [b4, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0006","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"296f5f388489c0a224ff329763648e558fbd35f469f54a58620c831d7dc38d81","family_id":"qf-mod47-mle-law-likelihood","seed":202611164006,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x5\",\"y5\"],\"observed\":\"x5\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a5\",\"b5\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a5, b5], observations=[x5, y5] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x5. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a5, b5], [x5, y5], [[3/4, 1/4], [1/4, 3/4]], x5, [[a5, 3/4], [b5, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a5, b5], [x5, y5], [[3/4, 1/4], [1/4, 3/4]], x5, [[a5, 3/4], [b5, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0007","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08ca62106ec5e8685c8a3a35c25118d38ecda544e0e05aae3193d0bfd478e604","family_id":"qf-mod47-mle-law-likelihood","seed":202611164007,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x6\",\"y6\"],\"observed\":\"x6\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a6\",\"b6\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a6, b6], observations=[x6, y6] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x6. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a6, b6], [x6, y6], [[3/4, 1/4], [1/4, 3/4]], x6, [[a6, 3/4], [b6, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a6, b6], [x6, y6], [[3/4, 1/4], [1/4, 3/4]], x6, [[a6, 3/4], [b6, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0008","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27f541fbe20683e5bb5a44884b1d7ade831ad5f0ddc1de20384dc9e6f8612325","family_id":"qf-mod47-mle-law-likelihood","seed":202611164008,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x7\",\"y7\"],\"observed\":\"x7\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a7\",\"b7\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a7, b7], observations=[x7, y7] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x7. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a7, b7], [x7, y7], [[3/4, 1/4], [1/4, 3/4]], x7, [[a7, 3/4], [b7, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a7, b7], [x7, y7], [[3/4, 1/4], [1/4, 3/4]], x7, [[a7, 3/4], [b7, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0009","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b27a7363445270c2ab477d905c16391207355c49c63f1a4deabe3ab789caf87","family_id":"qf-mod47-mle-law-likelihood","seed":202611164009,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x8\",\"y8\"],\"observed\":\"x8\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a8\",\"b8\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a8, b8], observations=[x8, y8] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x8. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a8, b8], [x8, y8], [[3/4, 1/4], [1/4, 3/4]], x8, [[a8, 3/4], [b8, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a8, b8], [x8, y8], [[3/4, 1/4], [1/4, 3/4]], x8, [[a8, 3/4], [b8, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0010","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc94306b7e256a0bc09a884b00a6f50902b15d0b2ba82b80a6614614922321a1","family_id":"qf-mod47-mle-law-likelihood","seed":202611164010,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x9\",\"y9\"],\"observed\":\"x9\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a9\",\"b9\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a9, b9], observations=[x9, y9] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x9. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a9, b9], [x9, y9], [[3/4, 1/4], [1/4, 3/4]], x9, [[a9, 3/4], [b9, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a9, b9], [x9, y9], [[3/4, 1/4], [1/4, 3/4]], x9, [[a9, 3/4], [b9, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0011","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3a7e26bd805333fca2b745d7f5a8f761db1260c7da3b73325d289ebb023e0bf","family_id":"qf-mod47-mle-law-likelihood","seed":202611164011,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x10\",\"y10\"],\"observed\":\"x10\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a10\",\"b10\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a10, b10], observations=[x10, y10] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x10. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a10, b10], [x10, y10], [[3/4, 1/4], [1/4, 3/4]], x10, [[a10, 3/4], [b10, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a10, b10], [x10, y10], [[3/4, 1/4], [1/4, 3/4]], x10, [[a10, 3/4], [b10, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0012","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"647e77292e5fa3ada59fa4a230fafdc030fdf4ca3967e6659c3664fd6ad2f7ae","family_id":"qf-mod47-mle-law-likelihood","seed":202611164012,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x11\",\"y11\"],\"observed\":\"x11\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a11\",\"b11\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a11, b11], observations=[x11, y11] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x11. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a11, b11], [x11, y11], [[3/4, 1/4], [1/4, 3/4]], x11, [[a11, 3/4], [b11, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a11, b11], [x11, y11], [[3/4, 1/4], [1/4, 3/4]], x11, [[a11, 3/4], [b11, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0013","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c02de93c638c71e65161d6ccd0ff86d73b9e5c549cecbd0e5aaea2974e264ae1","family_id":"qf-mod47-mle-law-likelihood","seed":202611164013,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x12\",\"y12\"],\"observed\":\"x12\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a12\",\"b12\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a12, b12], observations=[x12, y12] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x12. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a12, b12], [x12, y12], [[3/4, 1/4], [1/4, 3/4]], x12, [[a12, 3/4], [b12, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a12, b12], [x12, y12], [[3/4, 1/4], [1/4, 3/4]], x12, [[a12, 3/4], [b12, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0014","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"869fdd18b9954059e39aa3a7d60611b0f17a6b91e8b87d179d693e033b08f86e","family_id":"qf-mod47-mle-law-likelihood","seed":202611164014,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x13\",\"y13\"],\"observed\":\"x13\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a13\",\"b13\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a13, b13], observations=[x13, y13] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x13. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a13, b13], [x13, y13], [[3/4, 1/4], [1/4, 3/4]], x13, [[a13, 3/4], [b13, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a13, b13], [x13, y13], [[3/4, 1/4], [1/4, 3/4]], x13, [[a13, 3/4], [b13, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0015","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcd84a4ea205502db45fa7e70cbc4ab5ce18391321b365a4cad427e35f2fe38d","family_id":"qf-mod47-mle-law-likelihood","seed":202611164015,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x14\",\"y14\"],\"observed\":\"x14\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a14\",\"b14\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a14, b14], observations=[x14, y14] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x14. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a14, b14], [x14, y14], [[3/4, 1/4], [1/4, 3/4]], x14, [[a14, 3/4], [b14, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a14, b14], [x14, y14], [[3/4, 1/4], [1/4, 3/4]], x14, [[a14, 3/4], [b14, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0016","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"476be96028e7472fd89373f6126c1522d102ca8824bb3b88aaac7650fb28caa5","family_id":"qf-mod47-mle-law-likelihood","seed":202611164016,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x15\",\"y15\"],\"observed\":\"x15\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a15\",\"b15\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a15, b15], observations=[x15, y15] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x15. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a15, b15], [x15, y15], [[3/4, 1/4], [1/4, 3/4]], x15, [[a15, 3/4], [b15, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a15, b15], [x15, y15], [[3/4, 1/4], [1/4, 3/4]], x15, [[a15, 3/4], [b15, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0017","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83a4330f4cfe109eb31a29e307e1385cf5e5b0dd011bbb8db7ce58d95ed7dff3","family_id":"qf-mod47-mle-law-likelihood","seed":202611164017,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x16\",\"y16\"],\"observed\":\"x16\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a16\",\"b16\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a16, b16], observations=[x16, y16] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x16. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a16, b16], [x16, y16], [[3/4, 1/4], [1/4, 3/4]], x16, [[a16, 3/4], [b16, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a16, b16], [x16, y16], [[3/4, 1/4], [1/4, 3/4]], x16, [[a16, 3/4], [b16, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0018","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b751c6989ef2235dedfcf87c8e386e3698f41052b7c5954151c5e850cec3b0b","family_id":"qf-mod47-mle-law-likelihood","seed":202611164018,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x17\",\"y17\"],\"observed\":\"x17\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a17\",\"b17\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a17, b17], observations=[x17, y17] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x17. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a17, b17], [x17, y17], [[3/4, 1/4], [1/4, 3/4]], x17, [[a17, 3/4], [b17, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a17, b17], [x17, y17], [[3/4, 1/4], [1/4, 3/4]], x17, [[a17, 3/4], [b17, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0019","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbfbeeb29482ee802b2c213f7ecec92b7dc25d5e94abcb3b5702ddfce56d8216","family_id":"qf-mod47-mle-law-likelihood","seed":202611164019,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x18\",\"y18\"],\"observed\":\"x18\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a18\",\"b18\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a18, b18], observations=[x18, y18] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x18. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a18, b18], [x18, y18], [[3/4, 1/4], [1/4, 3/4]], x18, [[a18, 3/4], [b18, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a18, b18], [x18, y18], [[3/4, 1/4], [1/4, 3/4]], x18, [[a18, 3/4], [b18, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0020","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"484b4a2a047e44ea8ca8f16011d50c3f67e028f224eb8ce6ac81f37954cdc918","family_id":"qf-mod47-mle-law-likelihood","seed":202611164020,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x19\",\"y19\"],\"observed\":\"x19\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a19\",\"b19\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a19, b19], observations=[x19, y19] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x19. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a19, b19], [x19, y19], [[3/4, 1/4], [1/4, 3/4]], x19, [[a19, 3/4], [b19, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a19, b19], [x19, y19], [[3/4, 1/4], [1/4, 3/4]], x19, [[a19, 3/4], [b19, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0021","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a276dc4968b8b4ec93ebfc3c174b7b188753b669463d24218c24bb060d6b5db7","family_id":"qf-mod47-mle-law-likelihood","seed":202611164021,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x20\",\"y20\"],\"observed\":\"x20\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a20\",\"b20\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a20, b20], observations=[x20, y20] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x20. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a20, b20], [x20, y20], [[3/4, 1/4], [1/4, 3/4]], x20, [[a20, 3/4], [b20, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a20, b20], [x20, y20], [[3/4, 1/4], [1/4, 3/4]], x20, [[a20, 3/4], [b20, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0022","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88f3daf54d8f18b9387c244edfc7817e8299b254c46510a276976a133c09dbaa","family_id":"qf-mod47-mle-law-likelihood","seed":202611164022,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x21\",\"y21\"],\"observed\":\"x21\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a21\",\"b21\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a21, b21], observations=[x21, y21] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x21. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a21, b21], [x21, y21], [[3/4, 1/4], [1/4, 3/4]], x21, [[a21, 3/4], [b21, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a21, b21], [x21, y21], [[3/4, 1/4], [1/4, 3/4]], x21, [[a21, 3/4], [b21, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0023","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42ff93104c02470a2a61c417d630825a5e89b813495787333e6e7208118795d1","family_id":"qf-mod47-mle-law-likelihood","seed":202611164023,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x22\",\"y22\"],\"observed\":\"x22\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a22\",\"b22\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a22, b22], observations=[x22, y22] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x22. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a22, b22], [x22, y22], [[3/4, 1/4], [1/4, 3/4]], x22, [[a22, 3/4], [b22, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a22, b22], [x22, y22], [[3/4, 1/4], [1/4, 3/4]], x22, [[a22, 3/4], [b22, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0024","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7f5a11ce9983fe1bb9779227c1d12649024e77b1a9bb2888f9b8a23601819c1","family_id":"qf-mod47-mle-law-likelihood","seed":202611164024,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x23\",\"y23\"],\"observed\":\"x23\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a23\",\"b23\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a23, b23], observations=[x23, y23] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x23. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a23, b23], [x23, y23], [[3/4, 1/4], [1/4, 3/4]], x23, [[a23, 3/4], [b23, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a23, b23], [x23, y23], [[3/4, 1/4], [1/4, 3/4]], x23, [[a23, 3/4], [b23, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0025","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edaba4683fb480bd52d1534e758d7fe238b0a31f85711ab311eb07dd22853f27","family_id":"qf-mod47-mle-law-likelihood","seed":202611164025,"semantic_template":"qf-mod47-mle-law-likelihood/{\"observations\":[\"x24\",\"y24\"],\"observed\":\"x24\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a24\",\"b24\"]}"},"fields":{"title":"The Observed Likelihood Uses Every Qualified Parameter Law","stem":"Use complete theta labels=[a24, b24], observations=[x24, y24] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x24. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).","answer":{"kind":"expression","value":"([a24, b24], [x24, y24], [[3/4, 1/4], [1/4, 3/4]], x24, [[a24, 3/4], [b24, 1/4]], complete_normalized_finite_model_likelihood_table)","explanation":"Require the observation to lie in the carrier, then retain every theta likelihood from its aligned probability column. Likelihood is not a posterior distribution and does not introduce a prior or require summing to1 across theta. The exact result, retaining the requested order, is ([a24, b24], [x24, y24], [[3/4, 1/4], [1/4, 3/4]], x24, [[a24, 3/4], [b24, 1/4]], complete_normalized_finite_model_likelihood_table)."},"skills":["The Observed Likelihood Uses Every Qualified Parameter Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0026","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b766a05f73bd4f8b28147465326bbe5c1ae9b888f6c281084c0c1246d9b15b3d","family_id":"qf-mod47-mle-argmax","seed":202611164026,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x0\",\"y0\"],\"observed\":\"x0\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a0\",\"b0\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a0, b0], observations=[x0, y0] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x0. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x0, [[a0, 3/4], [b0, 1/4]], 3/4, [a0], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x0, [[a0, 3/4], [b0, 1/4]], 3/4, [a0], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0027","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb2ab30d2985161d50926eba817c4ddb1f0b33776bd5d810c3114b2da9753575","family_id":"qf-mod47-mle-argmax","seed":202611164027,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x1\",\"y1\"],\"observed\":\"x1\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a1\",\"b1\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a1, b1], observations=[x1, y1] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x1. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x1, [[a1, 3/4], [b1, 1/4]], 3/4, [a1], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x1, [[a1, 3/4], [b1, 1/4]], 3/4, [a1], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0028","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc0143142383544c01815d34ede3139ce55f4e37698f2e359390c6bb5eed81c3","family_id":"qf-mod47-mle-argmax","seed":202611164028,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x2\",\"y2\"],\"observed\":\"x2\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a2\",\"b2\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a2, b2], observations=[x2, y2] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x2. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x2, [[a2, 3/4], [b2, 1/4]], 3/4, [a2], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x2, [[a2, 3/4], [b2, 1/4]], 3/4, [a2], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0029","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35563b8a99870b6999742853e727a07d9cc4a3026677a6055adb81e5e0fe1ee5","family_id":"qf-mod47-mle-argmax","seed":202611164029,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x3\",\"y3\"],\"observed\":\"x3\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a3\",\"b3\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a3, b3], observations=[x3, y3] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x3. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x3, [[a3, 3/4], [b3, 1/4]], 3/4, [a3], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x3, [[a3, 3/4], [b3, 1/4]], 3/4, [a3], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0030","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a8277f79e253a95f42807134f7dcc6d54e10ef65028d3b03dc120d4b8b0ecfb","family_id":"qf-mod47-mle-argmax","seed":202611164030,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x4\",\"y4\"],\"observed\":\"x4\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a4\",\"b4\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a4, b4], observations=[x4, y4] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x4. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x4, [[a4, 3/4], [b4, 1/4]], 3/4, [a4], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x4, [[a4, 3/4], [b4, 1/4]], 3/4, [a4], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0031","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9a32d07b51e26e0d8a7fdf3c33283f977e6ec2178ccb891cc40b622e3a0688b","family_id":"qf-mod47-mle-argmax","seed":202611164031,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x5\",\"y5\"],\"observed\":\"x5\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a5\",\"b5\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a5, b5], observations=[x5, y5] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x5. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x5, [[a5, 3/4], [b5, 1/4]], 3/4, [a5], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x5, [[a5, 3/4], [b5, 1/4]], 3/4, [a5], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0032","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7aafb21a68ebe42c91e4a7390ef3ab3fbb4c86388195b0e95edc8896a700fa94","family_id":"qf-mod47-mle-argmax","seed":202611164032,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x6\",\"y6\"],\"observed\":\"x6\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a6\",\"b6\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a6, b6], observations=[x6, y6] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x6. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x6, [[a6, 3/4], [b6, 1/4]], 3/4, [a6], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x6, [[a6, 3/4], [b6, 1/4]], 3/4, [a6], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0033","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e6c767e9300166bd99a72b69fe4994f541f17fa11519bbc81cf55d4d6f74364","family_id":"qf-mod47-mle-argmax","seed":202611164033,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x7\",\"y7\"],\"observed\":\"x7\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a7\",\"b7\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a7, b7], observations=[x7, y7] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x7. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x7, [[a7, 3/4], [b7, 1/4]], 3/4, [a7], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x7, [[a7, 3/4], [b7, 1/4]], 3/4, [a7], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0034","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c535eea2d47a0b195fe312dcd1c81482040fd8493fe56ca81b93d4f95118d278","family_id":"qf-mod47-mle-argmax","seed":202611164034,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x8\",\"y8\"],\"observed\":\"x8\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a8\",\"b8\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a8, b8], observations=[x8, y8] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x8. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x8, [[a8, 3/4], [b8, 1/4]], 3/4, [a8], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x8, [[a8, 3/4], [b8, 1/4]], 3/4, [a8], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0035","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"574384a428d296ac7e6b22cf12e68f91ece89b081e0eb91fc382c8f90f6cd894","family_id":"qf-mod47-mle-argmax","seed":202611164035,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x9\",\"y9\"],\"observed\":\"x9\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a9\",\"b9\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a9, b9], observations=[x9, y9] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x9. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x9, [[a9, 3/4], [b9, 1/4]], 3/4, [a9], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x9, [[a9, 3/4], [b9, 1/4]], 3/4, [a9], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0036","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2606a9ba6d6beae91127df835a5b51c153b925de819013d6b7abc814c3f7423","family_id":"qf-mod47-mle-argmax","seed":202611164036,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x10\",\"y10\"],\"observed\":\"x10\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a10\",\"b10\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a10, b10], observations=[x10, y10] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x10. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x10, [[a10, 3/4], [b10, 1/4]], 3/4, [a10], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x10, [[a10, 3/4], [b10, 1/4]], 3/4, [a10], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0037","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f1263fefe5ce03bc8872ea3572a7e3702447c895f3a01b56308425686953eae","family_id":"qf-mod47-mle-argmax","seed":202611164037,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x11\",\"y11\"],\"observed\":\"x11\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a11\",\"b11\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a11, b11], observations=[x11, y11] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x11. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x11, [[a11, 3/4], [b11, 1/4]], 3/4, [a11], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x11, [[a11, 3/4], [b11, 1/4]], 3/4, [a11], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0038","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc49f05109c3f6ab826d605f01fc57405bceeff2b16ef2dbd9ee84946af65095","family_id":"qf-mod47-mle-argmax","seed":202611164038,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x12\",\"y12\"],\"observed\":\"x12\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a12\",\"b12\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a12, b12], observations=[x12, y12] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x12. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x12, [[a12, 3/4], [b12, 1/4]], 3/4, [a12], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x12, [[a12, 3/4], [b12, 1/4]], 3/4, [a12], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0039","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dc09093d08761c80665216f2bfb5af72faa48f0eb9dd3e7973daa017cee910c","family_id":"qf-mod47-mle-argmax","seed":202611164039,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x13\",\"y13\"],\"observed\":\"x13\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a13\",\"b13\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a13, b13], observations=[x13, y13] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x13. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x13, [[a13, 3/4], [b13, 1/4]], 3/4, [a13], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x13, [[a13, 3/4], [b13, 1/4]], 3/4, [a13], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0040","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46e9e6aab91756ef757fac01ca6a09da39e5a07387cfa76af375fcbf6b75ac59","family_id":"qf-mod47-mle-argmax","seed":202611164040,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x14\",\"y14\"],\"observed\":\"x14\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a14\",\"b14\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a14, b14], observations=[x14, y14] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x14. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x14, [[a14, 3/4], [b14, 1/4]], 3/4, [a14], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x14, [[a14, 3/4], [b14, 1/4]], 3/4, [a14], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0041","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3595fa7a17cfcbc5dde162463ec9965b8e184b9a94fb89d415dd3034776c5658","family_id":"qf-mod47-mle-argmax","seed":202611164041,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x15\",\"y15\"],\"observed\":\"x15\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a15\",\"b15\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a15, b15], observations=[x15, y15] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x15. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x15, [[a15, 3/4], [b15, 1/4]], 3/4, [a15], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x15, [[a15, 3/4], [b15, 1/4]], 3/4, [a15], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0042","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"290118ed1ead027a1007a4c278a0abf21a3579fb45773461efd486d004953234","family_id":"qf-mod47-mle-argmax","seed":202611164042,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x16\",\"y16\"],\"observed\":\"x16\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a16\",\"b16\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a16, b16], observations=[x16, y16] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x16. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x16, [[a16, 3/4], [b16, 1/4]], 3/4, [a16], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x16, [[a16, 3/4], [b16, 1/4]], 3/4, [a16], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0043","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a04fdbeecf17132fbe61ad5a3522d5a8e95594a4486a1b12efc7fb489e0a6ff","family_id":"qf-mod47-mle-argmax","seed":202611164043,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x17\",\"y17\"],\"observed\":\"x17\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a17\",\"b17\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a17, b17], observations=[x17, y17] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x17. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x17, [[a17, 3/4], [b17, 1/4]], 3/4, [a17], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x17, [[a17, 3/4], [b17, 1/4]], 3/4, [a17], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0044","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4698c1ce02a3428867ee8897a1ddb0fdcb617e0c8279bc0b4ba1e83505e3c930","family_id":"qf-mod47-mle-argmax","seed":202611164044,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x18\",\"y18\"],\"observed\":\"x18\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a18\",\"b18\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a18, b18], observations=[x18, y18] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x18. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x18, [[a18, 3/4], [b18, 1/4]], 3/4, [a18], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x18, [[a18, 3/4], [b18, 1/4]], 3/4, [a18], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0045","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52c6476c5e2838c270476747f56ac21d4e4558adcf2a51807fcdd83f2a58acb2","family_id":"qf-mod47-mle-argmax","seed":202611164045,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x19\",\"y19\"],\"observed\":\"x19\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a19\",\"b19\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a19, b19], observations=[x19, y19] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x19. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x19, [[a19, 3/4], [b19, 1/4]], 3/4, [a19], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x19, [[a19, 3/4], [b19, 1/4]], 3/4, [a19], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0046","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c512b7d0238524ff299e8851b975d22dd58c05c4a27f8fedcb09f1d282573375","family_id":"qf-mod47-mle-argmax","seed":202611164046,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x20\",\"y20\"],\"observed\":\"x20\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a20\",\"b20\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a20, b20], observations=[x20, y20] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x20. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x20, [[a20, 3/4], [b20, 1/4]], 3/4, [a20], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x20, [[a20, 3/4], [b20, 1/4]], 3/4, [a20], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0047","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff520dc0c172850f41c1400781f7c009c528f31b8794d234d503565d4ad02a32","family_id":"qf-mod47-mle-argmax","seed":202611164047,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x21\",\"y21\"],\"observed\":\"x21\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a21\",\"b21\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a21, b21], observations=[x21, y21] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x21. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x21, [[a21, 3/4], [b21, 1/4]], 3/4, [a21], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x21, [[a21, 3/4], [b21, 1/4]], 3/4, [a21], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0048","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ebe54efe970a82c4186c067d8eb27189c22f4aa876497006f1ca712065474e3","family_id":"qf-mod47-mle-argmax","seed":202611164048,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x22\",\"y22\"],\"observed\":\"x22\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a22\",\"b22\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a22, b22], observations=[x22, y22] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x22. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x22, [[a22, 3/4], [b22, 1/4]], 3/4, [a22], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x22, [[a22, 3/4], [b22, 1/4]], 3/4, [a22], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0049","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"032308e1870846513d72fea135dc420a26c568c0280b26a2496bed9d7e737f5b","family_id":"qf-mod47-mle-argmax","seed":202611164049,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x23\",\"y23\"],\"observed\":\"x23\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a23\",\"b23\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a23, b23], observations=[x23, y23] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x23. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x23, [[a23, 3/4], [b23, 1/4]], 3/4, [a23], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x23, [[a23, 3/4], [b23, 1/4]], 3/4, [a23], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0050","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b480d869010ee0a133d8f54f905a5280087ab4330ae6bb0633485ac49e153531","family_id":"qf-mod47-mle-argmax","seed":202611164050,"semantic_template":"qf-mod47-mle-argmax/{\"observations\":[\"x24\",\"y24\"],\"observed\":\"x24\",\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a24\",\"b24\"]}"},"fields":{"title":"Retain Every Likelihood Maximizer before Discussing Uniqueness","stem":"Use complete theta labels=[a24, b24], observations=[x24, y24] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x24. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).","answer":{"kind":"expression","value":"(x24, [[a24, 3/4], [b24, 1/4]], 3/4, [a24], Yes, MLE_argmax_need_not_be_unique)","explanation":"Compare the observation likelihood across all theta and return the full argmax set plus its uniqueness flag. A maximum value alone does not identify every tied parameter; MLE does not automatically imply unbiasedness, consistency or asymptotic optimality. The exact result, retaining the requested order, is (x24, [[a24, 3/4], [b24, 1/4]], 3/4, [a24], Yes, MLE_argmax_need_not_be_unique)."},"skills":["Retain Every Likelihood Maximizer before Discussing Uniqueness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0051","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0d8bf4423d1d86b10a30f8415c79e9444e1dcf3f8a3558264e03af6a39c6168","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164051,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x0\",\"y0\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a0\",\"b0\"],\"tie_order\":[\"a0\",\"b0\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a0, b0], observations=[x0, y0] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a0, b0] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a0, b0], [[x0, [a0], a0], [y0, [b0], b0]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a0, b0], [[x0, [a0], a0], [y0, [b0], b0]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0052","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"559342b017fa94279e81192eb701baa643a83b7dd14aaab8be9ec680d53c7329","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164052,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x1\",\"y1\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a1\",\"b1\"],\"tie_order\":[\"a1\",\"b1\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a1, b1], observations=[x1, y1] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a1, b1] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a1, b1], [[x1, [a1], a1], [y1, [b1], b1]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a1, b1], [[x1, [a1], a1], [y1, [b1], b1]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0053","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ed3b07cf780e973964eb276910fea52e9402cb09f604c61e4039c4c40e73c71","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164053,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x2\",\"y2\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a2\",\"b2\"],\"tie_order\":[\"a2\",\"b2\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a2, b2], observations=[x2, y2] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a2, b2] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a2, b2], [[x2, [a2], a2], [y2, [b2], b2]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a2, b2], [[x2, [a2], a2], [y2, [b2], b2]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0054","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01bc7a6604a8847bf67648a9b792a2e9924825dec5a708359bdeb02e23bea641","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164054,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x3\",\"y3\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a3\",\"b3\"],\"tie_order\":[\"a3\",\"b3\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a3, b3], observations=[x3, y3] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a3, b3] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a3, b3], [[x3, [a3], a3], [y3, [b3], b3]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a3, b3], [[x3, [a3], a3], [y3, [b3], b3]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0055","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2d0f1d3eff84ab226ccdc32425ce366c34be2eb112847be9e58964d6a2f11f7","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164055,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x4\",\"y4\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a4\",\"b4\"],\"tie_order\":[\"a4\",\"b4\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a4, b4], observations=[x4, y4] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a4, b4] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a4, b4], [[x4, [a4], a4], [y4, [b4], b4]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a4, b4], [[x4, [a4], a4], [y4, [b4], b4]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0056","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20ad636f3261540229416a05d3074dfba9615b401804cf8e80212cab1c9da0f9","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164056,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x5\",\"y5\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a5\",\"b5\"],\"tie_order\":[\"a5\",\"b5\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a5, b5], observations=[x5, y5] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a5, b5] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a5, b5], [[x5, [a5], a5], [y5, [b5], b5]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a5, b5], [[x5, [a5], a5], [y5, [b5], b5]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0057","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"365526b3a48784a4607093b2899c6661c53fe85fc52791c7ba4270fddcd3da25","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164057,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x6\",\"y6\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a6\",\"b6\"],\"tie_order\":[\"a6\",\"b6\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a6, b6], observations=[x6, y6] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a6, b6] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a6, b6], [[x6, [a6], a6], [y6, [b6], b6]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a6, b6], [[x6, [a6], a6], [y6, [b6], b6]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0058","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ebbae70a947c33505d17c376f125473032c654789db2549fbe67a85b70f0e03","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164058,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x7\",\"y7\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a7\",\"b7\"],\"tie_order\":[\"a7\",\"b7\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a7, b7], observations=[x7, y7] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a7, b7] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a7, b7], [[x7, [a7], a7], [y7, [b7], b7]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a7, b7], [[x7, [a7], a7], [y7, [b7], b7]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0059","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9540276759ad4bfe49f9761a1bdddf09e642a871a3ddac5840929b7b37a8f985","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164059,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x8\",\"y8\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a8\",\"b8\"],\"tie_order\":[\"a8\",\"b8\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a8, b8], observations=[x8, y8] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a8, b8] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a8, b8], [[x8, [a8], a8], [y8, [b8], b8]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a8, b8], [[x8, [a8], a8], [y8, [b8], b8]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0060","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c7792ce0e3bdb24ace55a8b0120c64256802b0aa61c3fbfc4e58b59c9cf0eb9","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164060,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x9\",\"y9\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a9\",\"b9\"],\"tie_order\":[\"a9\",\"b9\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a9, b9], observations=[x9, y9] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a9, b9] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a9, b9], [[x9, [a9], a9], [y9, [b9], b9]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a9, b9], [[x9, [a9], a9], [y9, [b9], b9]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0061","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfba572293185742d75ce8517b5f5314e30323c19f1f78d5b6fd7f73cc69a015","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164061,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x10\",\"y10\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a10\",\"b10\"],\"tie_order\":[\"a10\",\"b10\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a10, b10], observations=[x10, y10] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a10, b10] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a10, b10], [[x10, [a10], a10], [y10, [b10], b10]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a10, b10], [[x10, [a10], a10], [y10, [b10], b10]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0062","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d67f9a3d11e7e318a1f30c1c19fd9e5a6a075316536f02a77a81105be4baba2","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164062,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x11\",\"y11\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a11\",\"b11\"],\"tie_order\":[\"a11\",\"b11\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a11, b11], observations=[x11, y11] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a11, b11] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a11, b11], [[x11, [a11], a11], [y11, [b11], b11]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a11, b11], [[x11, [a11], a11], [y11, [b11], b11]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0063","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b28fe564abb8b0dea73ac7970fee722cea82a8d2528346ccb9b181568a238cb","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164063,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x12\",\"y12\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a12\",\"b12\"],\"tie_order\":[\"a12\",\"b12\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a12, b12], observations=[x12, y12] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a12, b12] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a12, b12], [[x12, [a12], a12], [y12, [b12], b12]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a12, b12], [[x12, [a12], a12], [y12, [b12], b12]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0064","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad931f212d6aa5ff8a533ef3f53d9703de3ea658939b4bbd8ac63a21885c9fde","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164064,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x13\",\"y13\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a13\",\"b13\"],\"tie_order\":[\"a13\",\"b13\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a13, b13], observations=[x13, y13] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a13, b13] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a13, b13], [[x13, [a13], a13], [y13, [b13], b13]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a13, b13], [[x13, [a13], a13], [y13, [b13], b13]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0065","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"379dcb78c2e2c920737252e9a6ee172883cf75f3a2c7dcd0a845f58819511646","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164065,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x14\",\"y14\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a14\",\"b14\"],\"tie_order\":[\"a14\",\"b14\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a14, b14], observations=[x14, y14] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a14, b14] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a14, b14], [[x14, [a14], a14], [y14, [b14], b14]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a14, b14], [[x14, [a14], a14], [y14, [b14], b14]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0066","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30dee81adb7e5d863aea88f3afbf463b654b39300ed1f738a5c20916acb420b3","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164066,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x15\",\"y15\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a15\",\"b15\"],\"tie_order\":[\"a15\",\"b15\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a15, b15], observations=[x15, y15] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a15, b15] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a15, b15], [[x15, [a15], a15], [y15, [b15], b15]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a15, b15], [[x15, [a15], a15], [y15, [b15], b15]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0067","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"070c993c45e481a6d127548a8ef4bc317a94626e23451e8ae57a5a5fff8d9465","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164067,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x16\",\"y16\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a16\",\"b16\"],\"tie_order\":[\"a16\",\"b16\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a16, b16], observations=[x16, y16] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a16, b16] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a16, b16], [[x16, [a16], a16], [y16, [b16], b16]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a16, b16], [[x16, [a16], a16], [y16, [b16], b16]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0068","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4811bee8ca57bf9fff48c0aadfe983ee303bd2c542eb4d74acc1a06a0e02363a","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164068,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x17\",\"y17\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a17\",\"b17\"],\"tie_order\":[\"a17\",\"b17\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a17, b17], observations=[x17, y17] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a17, b17] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a17, b17], [[x17, [a17], a17], [y17, [b17], b17]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a17, b17], [[x17, [a17], a17], [y17, [b17], b17]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0069","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13df5eef94260b9fc6e21688cf708c64ed7c83e485b86ca8e025834d1da41334","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164069,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x18\",\"y18\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a18\",\"b18\"],\"tie_order\":[\"a18\",\"b18\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a18, b18], observations=[x18, y18] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a18, b18] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a18, b18], [[x18, [a18], a18], [y18, [b18], b18]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a18, b18], [[x18, [a18], a18], [y18, [b18], b18]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0070","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"326a0f09e2f8808ee4385dd2d214243f9ec81ebda21d02abfa12215dc7ce04e5","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164070,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x19\",\"y19\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a19\",\"b19\"],\"tie_order\":[\"a19\",\"b19\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a19, b19], observations=[x19, y19] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a19, b19] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a19, b19], [[x19, [a19], a19], [y19, [b19], b19]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a19, b19], [[x19, [a19], a19], [y19, [b19], b19]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0071","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a07aceb95305f253cfe8afffb890d14cfa68b5a56bb75d788d3da9e2ddbda052","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164071,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x20\",\"y20\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a20\",\"b20\"],\"tie_order\":[\"a20\",\"b20\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a20, b20], observations=[x20, y20] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a20, b20] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a20, b20], [[x20, [a20], a20], [y20, [b20], b20]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a20, b20], [[x20, [a20], a20], [y20, [b20], b20]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0072","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"012bb06e6cfd15883404e58a598cf6f8c421fd9f4a9a43994cf8b5320000ea7d","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164072,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x21\",\"y21\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a21\",\"b21\"],\"tie_order\":[\"a21\",\"b21\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a21, b21], observations=[x21, y21] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a21, b21] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a21, b21], [[x21, [a21], a21], [y21, [b21], b21]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a21, b21], [[x21, [a21], a21], [y21, [b21], b21]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0073","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5a99729f7eed597515e3f87db994f269eefc406d3291156c11bc694ce73b1be","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164073,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x22\",\"y22\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a22\",\"b22\"],\"tie_order\":[\"a22\",\"b22\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a22, b22], observations=[x22, y22] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a22, b22] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a22, b22], [[x22, [a22], a22], [y22, [b22], b22]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a22, b22], [[x22, [a22], a22], [y22, [b22], b22]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0074","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d53dea9e04f9c3decbec8fa70ce0bd1ff54da40c31de3b7d9354dc4da64c98d","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164074,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x23\",\"y23\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a23\",\"b23\"],\"tie_order\":[\"a23\",\"b23\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a23, b23], observations=[x23, y23] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a23, b23] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a23, b23], [[x23, [a23], a23], [y23, [b23], b23]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a23, b23], [[x23, [a23], a23], [y23, [b23], b23]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0075","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fcba2c8a7fe13f6eac5d17184fe6a3bc1f2f796fccfabded782f3e35217ea7c","family_id":"qf-mod47-mle-total-tie-estimator","seed":202611164075,"semantic_template":"qf-mod47-mle-total-tie-estimator/{\"observations\":[\"x24\",\"y24\"],\"probability_rows\":[[[3,4],[1,4]],[[1,4],[3,4]]],\"theta_values\":[\"a24\",\"b24\"],\"tie_order\":[\"a24\",\"b24\"]}"},"fields":{"title":"One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator","stem":"Use complete theta labels=[a24, b24], observations=[x24, y24] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a24, b24] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).","answer":{"kind":"expression","value":"([a24, b24], [[x24, [a24], a24], [y24, [b24], b24]], Yes, theta_independent_tie_rule_makes_a_total_estimator)","explanation":"For every observation retain its complete argmax set and choose the first member in the same tie order. All-zero observation columns also have all parameters as maximizers. A fixed estimator value is not a posterior or an estimator-quality theorem. The exact result, retaining the requested order, is ([a24, b24], [[x24, [a24], a24], [y24, [b24], b24]], Yes, theta_independent_tie_rule_makes_a_total_estimator)."},"skills":["One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0076","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab1e8dfef316d92bbfc13c559f06ca9576b2cdb1848863264f6a58cda1efde47","family_id":"qf-mod47-mle-boundary","seed":202611164076,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x0\",\"y0\"],\"observed\":\"x0\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a0\",\"b0\"],\"tie_order\":[\"a0\",\"b0\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a0, b0], observations=[x0, y0] and aligned exact probability rows=[[0, 1], [0, 1]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x0, mode=all_zero_likelihood, fixed tie order=[a0, b0]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(all_zero_likelihood, x0, [[a0, 0], [b0, 0]], 0, [a0, b0], [a0, b0], [x0, [a0, b0], a0], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (all_zero_likelihood, x0, [[a0, 0], [b0, 0]], 0, [a0, b0], [a0, b0], [x0, [a0, b0], a0], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0077","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a78a4be1db13811d45226451dbbfaf0a8fa5637f7adcefc07da8cfc6be1d459b","family_id":"qf-mod47-mle-boundary","seed":202611164077,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x1\",\"y1\"],\"observed\":\"x1\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a1\",\"b1\"],\"tie_order\":[\"a1\",\"b1\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a1, b1], observations=[x1, y1] and aligned exact probability rows=[[[1, 2], [1, 2]], [[1, 2], [1, 2]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x1, mode=tie, fixed tie order=[a1, b1]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(tie, x1, [[a1, 1/2], [b1, 1/2]], 1/2, [a1, b1], [a1, b1], [x1, [a1, b1], a1], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (tie, x1, [[a1, 1/2], [b1, 1/2]], 1/2, [a1, b1], [a1, b1], [x1, [a1, b1], a1], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0078","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21972968102e860461ee67b1aa7f4de6bd0e8b69041c800603312e55a2c80cf5","family_id":"qf-mod47-mle-boundary","seed":202611164078,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x2\",\"y2\"],\"observed\":\"x2\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a2\",\"b2\"],\"tie_order\":[\"a2\",\"b2\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a2, b2], observations=[x2, y2] and aligned exact probability rows=[[0, 1], [0, 1]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x2, mode=all_zero_likelihood, fixed tie order=[a2, b2]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(all_zero_likelihood, x2, [[a2, 0], [b2, 0]], 0, [a2, b2], [a2, b2], [x2, [a2, b2], a2], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (all_zero_likelihood, x2, [[a2, 0], [b2, 0]], 0, [a2, b2], [a2, b2], [x2, [a2, b2], a2], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0079","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecaee93f1dc9c11ad53e09cd0229f412644b0025cbc07d6761a3e72f0826701f","family_id":"qf-mod47-mle-boundary","seed":202611164079,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x3\",\"y3\"],\"observed\":\"x3\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a3\",\"b3\"],\"tie_order\":[\"a3\",\"b3\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a3, b3], observations=[x3, y3] and aligned exact probability rows=[[[1, 2], [1, 2]], [[1, 2], [1, 2]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x3, mode=tie, fixed tie order=[a3, b3]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(tie, x3, [[a3, 1/2], [b3, 1/2]], 1/2, [a3, b3], [a3, b3], [x3, [a3, b3], a3], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (tie, x3, [[a3, 1/2], [b3, 1/2]], 1/2, [a3, b3], [a3, b3], [x3, [a3, b3], a3], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0080","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f43c02efb1580ba81351244bded8403fe34a7137e6c1db1193cd5e686b5234b2","family_id":"qf-mod47-mle-boundary","seed":202611164080,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x4\",\"y4\"],\"observed\":\"x4\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a4\",\"b4\"],\"tie_order\":[\"a4\",\"b4\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a4, b4], observations=[x4, y4] and aligned exact probability rows=[[0, 1], [0, 1]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x4, mode=all_zero_likelihood, fixed tie order=[a4, b4]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(all_zero_likelihood, x4, [[a4, 0], [b4, 0]], 0, [a4, b4], [a4, b4], [x4, [a4, b4], a4], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (all_zero_likelihood, x4, [[a4, 0], [b4, 0]], 0, [a4, b4], [a4, b4], [x4, [a4, b4], a4], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0081","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb01b60ad9aba57b3e79ecfd4442e2e11183c343a04c0fc1ef3a8fb8d90ec9da","family_id":"qf-mod47-mle-boundary","seed":202611164081,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x5\",\"y5\"],\"observed\":\"x5\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a5\",\"b5\"],\"tie_order\":[\"a5\",\"b5\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a5, b5], observations=[x5, y5] and aligned exact probability rows=[[[1, 2], [1, 2]], [[1, 2], [1, 2]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x5, mode=tie, fixed tie order=[a5, b5]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(tie, x5, [[a5, 1/2], [b5, 1/2]], 1/2, [a5, b5], [a5, b5], [x5, [a5, b5], a5], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (tie, x5, [[a5, 1/2], [b5, 1/2]], 1/2, [a5, b5], [a5, b5], [x5, [a5, b5], a5], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0082","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c98c6e695b9990830d0aef9e21cb76eae43bed18b29808c3a1b13871d4aa1e78","family_id":"qf-mod47-mle-boundary","seed":202611164082,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x6\",\"y6\"],\"observed\":\"x6\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a6\",\"b6\"],\"tie_order\":[\"a6\",\"b6\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a6, b6], observations=[x6, y6] and aligned exact probability rows=[[0, 1], [0, 1]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x6, mode=all_zero_likelihood, fixed tie order=[a6, b6]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(all_zero_likelihood, x6, [[a6, 0], [b6, 0]], 0, [a6, b6], [a6, b6], [x6, [a6, b6], a6], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (all_zero_likelihood, x6, [[a6, 0], [b6, 0]], 0, [a6, b6], [a6, b6], [x6, [a6, b6], a6], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0083","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e846c2771f8758087f48466dda9b1cd1169b08b66185f8fd37a49ccf1722a09","family_id":"qf-mod47-mle-boundary","seed":202611164083,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x7\",\"y7\"],\"observed\":\"x7\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a7\",\"b7\"],\"tie_order\":[\"a7\",\"b7\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a7, b7], observations=[x7, y7] and aligned exact probability rows=[[[1, 2], [1, 2]], [[1, 2], [1, 2]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x7, mode=tie, fixed tie order=[a7, b7]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(tie, x7, [[a7, 1/2], [b7, 1/2]], 1/2, [a7, b7], [a7, b7], [x7, [a7, b7], a7], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (tie, x7, [[a7, 1/2], [b7, 1/2]], 1/2, [a7, b7], [a7, b7], [x7, [a7, b7], a7], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0084","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"499fe9d5dd47f45681da3266eae1abefa8f3be8af02a9eddcc29b266550d5e08","family_id":"qf-mod47-mle-boundary","seed":202611164084,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x8\",\"y8\"],\"observed\":\"x8\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a8\",\"b8\"],\"tie_order\":[\"a8\",\"b8\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a8, b8], observations=[x8, y8] and aligned exact probability rows=[[0, 1], [0, 1]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x8, mode=all_zero_likelihood, fixed tie order=[a8, b8]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(all_zero_likelihood, x8, [[a8, 0], [b8, 0]], 0, [a8, b8], [a8, b8], [x8, [a8, b8], a8], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (all_zero_likelihood, x8, [[a8, 0], [b8, 0]], 0, [a8, b8], [a8, b8], [x8, [a8, b8], a8], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0085","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41075c2232feca81e6f9d0fba83dba216e15ee23ed38889a8d70ca871e2727af","family_id":"qf-mod47-mle-boundary","seed":202611164085,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x9\",\"y9\"],\"observed\":\"x9\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a9\",\"b9\"],\"tie_order\":[\"a9\",\"b9\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a9, b9], observations=[x9, y9] and aligned exact probability rows=[[[1, 2], [1, 2]], [[1, 2], [1, 2]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x9, mode=tie, fixed tie order=[a9, b9]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(tie, x9, [[a9, 1/2], [b9, 1/2]], 1/2, [a9, b9], [a9, b9], [x9, [a9, b9], a9], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (tie, x9, [[a9, 1/2], [b9, 1/2]], 1/2, [a9, b9], [a9, b9], [x9, [a9, b9], a9], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0086","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d59bfdbfbb2c9f0a7ebe6a53d0e0bbf5ccb40139782b0f6876668dba0524ea7","family_id":"qf-mod47-mle-boundary","seed":202611164086,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x10\",\"y10\"],\"observed\":\"x10\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a10\",\"b10\"],\"tie_order\":[\"a10\",\"b10\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a10, b10], observations=[x10, y10] and aligned exact probability rows=[[0, 1], [0, 1]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x10, mode=all_zero_likelihood, fixed tie order=[a10, b10]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(all_zero_likelihood, x10, [[a10, 0], [b10, 0]], 0, [a10, b10], [a10, b10], [x10, [a10, b10], a10], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (all_zero_likelihood, x10, [[a10, 0], [b10, 0]], 0, [a10, b10], [a10, b10], [x10, [a10, b10], a10], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0087","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c9941d97598d32ae6d233d4469fd8816ecea718836c7ecf9a8785209c5024cc","family_id":"qf-mod47-mle-boundary","seed":202611164087,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x11\",\"y11\"],\"observed\":\"x11\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a11\",\"b11\"],\"tie_order\":[\"a11\",\"b11\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a11, b11], observations=[x11, y11] and aligned exact probability rows=[[[1, 2], [1, 2]], [[1, 2], [1, 2]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x11, mode=tie, fixed tie order=[a11, b11]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(tie, x11, [[a11, 1/2], [b11, 1/2]], 1/2, [a11, b11], [a11, b11], [x11, [a11, b11], a11], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (tie, x11, [[a11, 1/2], [b11, 1/2]], 1/2, [a11, b11], [a11, b11], [x11, [a11, b11], a11], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0088","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21602080f6824b046b0caa5f914f7f93eab05e5bb187ac48971e71f8db2a9fdc","family_id":"qf-mod47-mle-boundary","seed":202611164088,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x12\",\"y12\"],\"observed\":\"x12\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a12\",\"b12\"],\"tie_order\":[\"a12\",\"b12\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a12, b12], observations=[x12, y12] and aligned exact probability rows=[[0, 1], [0, 1]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x12, mode=all_zero_likelihood, fixed tie order=[a12, b12]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(all_zero_likelihood, x12, [[a12, 0], [b12, 0]], 0, [a12, b12], [a12, b12], [x12, [a12, b12], a12], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (all_zero_likelihood, x12, [[a12, 0], [b12, 0]], 0, [a12, b12], [a12, b12], [x12, [a12, b12], a12], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0089","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a3a9f173a0f77720c5826945f7176d017f4d4df4418f17015e51e5d5a780bd0","family_id":"qf-mod47-mle-boundary","seed":202611164089,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x13\",\"y13\"],\"observed\":\"x13\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a13\",\"b13\"],\"tie_order\":[\"a13\",\"b13\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a13, b13], observations=[x13, y13] and aligned exact probability rows=[[[1, 2], [1, 2]], [[1, 2], [1, 2]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x13, mode=tie, fixed tie order=[a13, b13]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(tie, x13, [[a13, 1/2], [b13, 1/2]], 1/2, [a13, b13], [a13, b13], [x13, [a13, b13], a13], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (tie, x13, [[a13, 1/2], [b13, 1/2]], 1/2, [a13, b13], [a13, b13], [x13, [a13, b13], a13], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0090","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf50bb19f61f360a8ea6c37a0a1b627459da69eeb75a6b7a614eb06b610a9bb2","family_id":"qf-mod47-mle-boundary","seed":202611164090,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x14\",\"y14\"],\"observed\":\"x14\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a14\",\"b14\"],\"tie_order\":[\"a14\",\"b14\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a14, b14], observations=[x14, y14] and aligned exact probability rows=[[0, 1], [0, 1]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x14, mode=all_zero_likelihood, fixed tie order=[a14, b14]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(all_zero_likelihood, x14, [[a14, 0], [b14, 0]], 0, [a14, b14], [a14, b14], [x14, [a14, b14], a14], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (all_zero_likelihood, x14, [[a14, 0], [b14, 0]], 0, [a14, b14], [a14, b14], [x14, [a14, b14], a14], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0091","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4e14fcd257101f0e58a7264e77042496c52f6e3a080a738b48f31ac25c10660","family_id":"qf-mod47-mle-boundary","seed":202611164091,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x15\",\"y15\"],\"observed\":\"x15\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a15\",\"b15\"],\"tie_order\":[\"a15\",\"b15\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a15, b15], observations=[x15, y15] and aligned exact probability rows=[[[1, 2], [1, 2]], [[1, 2], [1, 2]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x15, mode=tie, fixed tie order=[a15, b15]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(tie, x15, [[a15, 1/2], [b15, 1/2]], 1/2, [a15, b15], [a15, b15], [x15, [a15, b15], a15], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (tie, x15, [[a15, 1/2], [b15, 1/2]], 1/2, [a15, b15], [a15, b15], [x15, [a15, b15], a15], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0092","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f3dde5d60682359db9a8dc6f80fe53ca020cffa26c612a566f1b9fe3fcacf36","family_id":"qf-mod47-mle-boundary","seed":202611164092,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x16\",\"y16\"],\"observed\":\"x16\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a16\",\"b16\"],\"tie_order\":[\"a16\",\"b16\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a16, b16], observations=[x16, y16] and aligned exact probability rows=[[0, 1], [0, 1]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x16, mode=all_zero_likelihood, fixed tie order=[a16, b16]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(all_zero_likelihood, x16, [[a16, 0], [b16, 0]], 0, [a16, b16], [a16, b16], [x16, [a16, b16], a16], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (all_zero_likelihood, x16, [[a16, 0], [b16, 0]], 0, [a16, b16], [a16, b16], [x16, [a16, b16], a16], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0093","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c29ed4a315aeeffee29529a72f37abdc93d37de077fe7bd392b1b8a51eb844a3","family_id":"qf-mod47-mle-boundary","seed":202611164093,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x17\",\"y17\"],\"observed\":\"x17\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a17\",\"b17\"],\"tie_order\":[\"a17\",\"b17\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a17, b17], observations=[x17, y17] and aligned exact probability rows=[[[1, 2], [1, 2]], [[1, 2], [1, 2]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x17, mode=tie, fixed tie order=[a17, b17]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(tie, x17, [[a17, 1/2], [b17, 1/2]], 1/2, [a17, b17], [a17, b17], [x17, [a17, b17], a17], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (tie, x17, [[a17, 1/2], [b17, 1/2]], 1/2, [a17, b17], [a17, b17], [x17, [a17, b17], a17], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0094","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06b70a4020602e4babe5407b94306ae4afc1dc1f19a5ab41a5fcb391e507a8b6","family_id":"qf-mod47-mle-boundary","seed":202611164094,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x18\",\"y18\"],\"observed\":\"x18\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a18\",\"b18\"],\"tie_order\":[\"a18\",\"b18\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a18, b18], observations=[x18, y18] and aligned exact probability rows=[[0, 1], [0, 1]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x18, mode=all_zero_likelihood, fixed tie order=[a18, b18]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(all_zero_likelihood, x18, [[a18, 0], [b18, 0]], 0, [a18, b18], [a18, b18], [x18, [a18, b18], a18], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (all_zero_likelihood, x18, [[a18, 0], [b18, 0]], 0, [a18, b18], [a18, b18], [x18, [a18, b18], a18], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0095","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4026a6e5c5406e5d3ff7604c0a73dc63f13a30435d386e5df02b410918c5c51d","family_id":"qf-mod47-mle-boundary","seed":202611164095,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x19\",\"y19\"],\"observed\":\"x19\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a19\",\"b19\"],\"tie_order\":[\"a19\",\"b19\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a19, b19], observations=[x19, y19] and aligned exact probability rows=[[[1, 2], [1, 2]], [[1, 2], [1, 2]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x19, mode=tie, fixed tie order=[a19, b19]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(tie, x19, [[a19, 1/2], [b19, 1/2]], 1/2, [a19, b19], [a19, b19], [x19, [a19, b19], a19], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (tie, x19, [[a19, 1/2], [b19, 1/2]], 1/2, [a19, b19], [a19, b19], [x19, [a19, b19], a19], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0096","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72f50b8d5dcb179bc8d70074e603a8cd254cf65a730f075402770ff98c16c37b","family_id":"qf-mod47-mle-boundary","seed":202611164096,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x20\",\"y20\"],\"observed\":\"x20\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a20\",\"b20\"],\"tie_order\":[\"a20\",\"b20\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a20, b20], observations=[x20, y20] and aligned exact probability rows=[[0, 1], [0, 1]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x20, mode=all_zero_likelihood, fixed tie order=[a20, b20]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(all_zero_likelihood, x20, [[a20, 0], [b20, 0]], 0, [a20, b20], [a20, b20], [x20, [a20, b20], a20], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (all_zero_likelihood, x20, [[a20, 0], [b20, 0]], 0, [a20, b20], [a20, b20], [x20, [a20, b20], a20], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0097","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53dc2bad061001226db43dd822b517e05eada447bfa25a7ff5c70a0217ca60aa","family_id":"qf-mod47-mle-boundary","seed":202611164097,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x21\",\"y21\"],\"observed\":\"x21\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a21\",\"b21\"],\"tie_order\":[\"a21\",\"b21\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a21, b21], observations=[x21, y21] and aligned exact probability rows=[[[1, 2], [1, 2]], [[1, 2], [1, 2]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x21, mode=tie, fixed tie order=[a21, b21]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(tie, x21, [[a21, 1/2], [b21, 1/2]], 1/2, [a21, b21], [a21, b21], [x21, [a21, b21], a21], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (tie, x21, [[a21, 1/2], [b21, 1/2]], 1/2, [a21, b21], [a21, b21], [x21, [a21, b21], a21], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0098","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"054e1a76807ca5c9739270dae686e5652fdaafc0f8c8a95840f2b32d3c373bf7","family_id":"qf-mod47-mle-boundary","seed":202611164098,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x22\",\"y22\"],\"observed\":\"x22\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a22\",\"b22\"],\"tie_order\":[\"a22\",\"b22\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a22, b22], observations=[x22, y22] and aligned exact probability rows=[[0, 1], [0, 1]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x22, mode=all_zero_likelihood, fixed tie order=[a22, b22]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(all_zero_likelihood, x22, [[a22, 0], [b22, 0]], 0, [a22, b22], [a22, b22], [x22, [a22, b22], a22], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (all_zero_likelihood, x22, [[a22, 0], [b22, 0]], 0, [a22, b22], [a22, b22], [x22, [a22, b22], a22], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0099","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4f3ab0bd050d27d3e21de3c5827e59cee8194e37c3ebab62ddc7a650f431955","family_id":"qf-mod47-mle-boundary","seed":202611164099,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"tie\",\"observations\":[\"x23\",\"y23\"],\"observed\":\"x23\",\"probability_rows\":[[[1,2],[1,2]],[[1,2],[1,2]]],\"theta_values\":[\"a23\",\"b23\"],\"tie_order\":[\"a23\",\"b23\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a23, b23], observations=[x23, y23] and aligned exact probability rows=[[[1, 2], [1, 2]], [[1, 2], [1, 2]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x23, mode=tie, fixed tie order=[a23, b23]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(tie, x23, [[a23, 1/2], [b23, 1/2]], 1/2, [a23, b23], [a23, b23], [x23, [a23, b23], a23], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (tie, x23, [[a23, 1/2], [b23, 1/2]], 1/2, [a23, b23], [a23, b23], [x23, [a23, b23], a23], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0100","batch_id":"batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1f85e3cc135020c8fef1ac7a4635cd1d32535e46e9d5555acb62aa28740347f","family_id":"qf-mod47-mle-boundary","seed":202611164100,"semantic_template":"qf-mod47-mle-boundary/{\"mode\":\"all_zero_likelihood\",\"observations\":[\"x24\",\"y24\"],\"observed\":\"x24\",\"probability_rows\":[[[0,1],[1,1]],[[0,1],[1,1]]],\"theta_values\":[\"a24\",\"b24\"],\"tie_order\":[\"a24\",\"b24\"]}"},"fields":{"title":"Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule","stem":"Use complete theta labels=[a24, b24], observations=[x24, y24] and aligned exact probability rows=[[0, 1], [0, 1]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x24, mode=all_zero_likelihood, fixed tie order=[a24, b24]. Give (mode, observed, likelihood_rows, maximum_likelihood, argmax_parameters, tie_order, selected_estimator_row, posterior_or_quality_claim, boundary).","answer":{"kind":"expression","value":"(all_zero_likelihood, x24, [[a24, 0], [b24, 0]], 0, [a24, b24], [a24, b24], [x24, [a24, b24], a24], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)","explanation":"Require a genuine multiple argmax for tie mode or a zero maximum for all_zero_likelihood. Retain all maximizers and the chosen estimator row. An event impossible under every parameter gives no positive likelihood evidence, but its zero argmax and tie-rule value are still defined by this finite convention. Do not invent a normalized posterior or an asymptotic conclusion. The exact result, retaining the requested order, is (all_zero_likelihood, x24, [[a24, 0], [b24, 0]], 0, [a24, b24], [a24, b24], [x24, [a24, b24], a24], No, MLE_maximizes_likelihood_not_posterior_and_does_not_imply_unbiased_consistent_or_asymptotically_optimal)."},"skills":["Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}