{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261114-r05-estimator-mean-squared-error-0082", "pilot_number": "R0049-25132", "number": 25132, "family_id": "qf-mod45-mse-observed-unbiased-boundary", "title": { "zh-Hant": "第 25132 題：估計量均方誤差", "en": "Question 25132: One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261114-r05-estimator-mean-squared-error-0082", "batch_id": "batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-estimator-mean-squared-error" ], "title_zh": "估計量均方誤差", "stem_zh": "scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":-6,\"support\":[-8,-6,-4],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(-6, -8, 4, -6, 是, 2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)", "explanation_zh": "Wave6第八層核對=(-6, -8, 4, -6, 是, 2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "估計量均方誤差" ] }, "variant": { "family_id": "qf-mod45-mse-observed-unbiased-boundary", "seed": 202611145082, "parameters": { "theta": -6, "support": [ -8, -6, -4 ], "probabilities": [ [ 1, 4 ], [ 1, 2 ], [ 1, 4 ] ], "observed": -8, "domain": "Exact_finite_scalar_estimator_MSE", "operation": "mse-observed-unbiased-boundary", "semantic_template": "qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-8,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-4,1],[-6,1],[-8,1]],\"theta\":[-6,1]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261114-r05-estimator-mean-squared-error-0082", "batch_id": "batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "d99d58f81fa1ef3efe20d71b9f921f426680debcc553d5e87fbba4b364fa7123", "family_id": "qf-mod45-mse-observed-unbiased-boundary", "seed": 202611145082, "semantic_template": "qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-8,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-4,1],[-6,1],[-8,1]],\"theta\":[-6,1]}" }, "fields": { "title": "One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency", "stem": "Use scalar theta=-6, complete listed estimator values=[-8, -6, -4] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-8. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).", "answer": { "kind": "expression", "value": "(-6, -8, 4, -6, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)", "explanation": "Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-6, -8, 4, -6, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)." }, "skills": [ "One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency" ], "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." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":-6,\"support\":[-8,-6,-4],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use scalar theta=-6, complete listed estimator values=[-8, -6, -4] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-8. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [mse-observed-unbiased-boundary]", "after": "derived = [[theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "本頁列完整來源欄序，不需JavaScript；LP三種mode雖同為6欄，角色仍由原mode選，不由答案形狀猜。", "en": "List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "MSE ORIGINAL single theta ORIGINAL probability cumulative0 := sum(0, 1/4)", "after": "MSE ORIGINAL single theta ORIGINAL probability cumulative0 = 1/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta ORIGINAL probability cumulative0，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta ORIGINAL probability cumulative0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "1/4" ], "output": "1/4", "label": "MSE ORIGINAL single theta ORIGINAL probability cumulative0" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "MSE ORIGINAL single theta ORIGINAL probability cumulative1 := sum(1/4, 1/2)", "after": "MSE ORIGINAL single theta ORIGINAL probability cumulative1 = 3/4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta ORIGINAL probability cumulative1，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta ORIGINAL probability cumulative1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1/4", "1/2" ], "output": "3/4", "label": "MSE ORIGINAL single theta ORIGINAL probability cumulative1" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "MSE ORIGINAL single theta ORIGINAL probability cumulative2 := sum(3/4, 1/4)", "after": "MSE ORIGINAL single theta ORIGINAL probability cumulative2 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta ORIGINAL probability cumulative2，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta ORIGINAL probability cumulative2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "3/4", "1/4" ], "output": "1", "label": "MSE ORIGINAL single theta ORIGINAL probability cumulative2" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [MSE ORIGINAL single theta, -6, [-8, -6, -4], [[1, 4], [1, 2], [1, 4]]]", "after": "derived = [[[-8, 1/4], [-6, 1/2], [-4, 1/4]], 1]", "relation": "implies", "rule_id": "MSE_same_theta_complete_law_numeric_sorted_ALIGNED_null_rows", "conditions": [], "explanation": { "zh-Hant": "本題固定同一P_theta；機率原序先核非負與總和1，再將value/mass一起按數值排。零mass列保留，不另重正規化，也不將T輸出當參數下標。", "en": "Fix this same P_theta.Check original-order nonnegative masses summing to1,then sort values numerically with their aligned masses.Keep null rows,without renormalizing or treating estimator values as parameter indices." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "MSE ORIGINAL single theta exact mean product0 := mul(-8, 1/4)", "after": "MSE ORIGINAL single theta exact mean product0 = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta exact mean product0，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta exact mean product0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-8", "1/4" ], "output": "-2", "label": "MSE ORIGINAL single theta exact mean product0" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "MSE ORIGINAL single theta exact mean cumulative0 := sum(0, -2)", "after": "MSE ORIGINAL single theta exact mean cumulative0 = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta exact mean cumulative0，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta exact mean cumulative0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "-2" ], "output": "-2", "label": "MSE ORIGINAL single theta exact mean cumulative0" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "MSE ORIGINAL single theta exact mean product1 := mul(-6, 1/2)", "after": "MSE ORIGINAL single theta exact mean product1 = -3", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta exact mean product1，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta exact mean product1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-6", "1/2" ], "output": "-3", "label": "MSE ORIGINAL single theta exact mean product1" } }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "MSE ORIGINAL single theta exact mean cumulative1 := sum(-2, -3)", "after": "MSE ORIGINAL single theta exact mean cumulative1 = -5", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta exact mean cumulative1，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta exact mean cumulative1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-2", "-3" ], "output": "-5", "label": "MSE ORIGINAL single theta exact mean cumulative1" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "MSE ORIGINAL single theta exact mean product2 := mul(-4, 1/4)", "after": "MSE ORIGINAL single theta exact mean product2 = -1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta exact mean product2，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta exact mean product2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-4", "1/4" ], "output": "-1", "label": "MSE ORIGINAL single theta exact mean product2" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "MSE ORIGINAL single theta exact mean cumulative2 := sum(-5, -1)", "after": "MSE ORIGINAL single theta exact mean cumulative2 = -6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta exact mean cumulative2，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta exact mean cumulative2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "-5", "-1" ], "output": "-6", "label": "MSE ORIGINAL single theta exact mean cumulative2" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "MSE ORIGINAL single theta error row0 := sub(-8, -6)", "after": "MSE ORIGINAL single theta error row0 = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta error row0，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta error row0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-8", "-6" ], "output": "-2", "label": "MSE ORIGINAL single theta error row0" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "MSE ORIGINAL single theta squared error row0 := mul(-2, -2)", "after": "MSE ORIGINAL single theta squared error row0 = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta squared error row0，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta squared error row0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-2" ], "output": "4", "label": "MSE ORIGINAL single theta squared error row0" } }, { "id": "S014", "uses": [ "G001", "S013" ], "before": "MSE ORIGINAL single theta weighted error row0 := mul(1/4, 4)", "after": "MSE ORIGINAL single theta weighted error row0 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta weighted error row0，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta weighted error row0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/4", "4" ], "output": "1", "label": "MSE ORIGINAL single theta weighted error row0" } }, { "id": "S015", "uses": [ "G001", "S014" ], "before": "MSE ORIGINAL single theta MSE cumulative0 := sum(0, 1)", "after": "MSE ORIGINAL single theta MSE cumulative0 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta MSE cumulative0，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta MSE cumulative0 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "0", "1" ], "output": "1", "label": "MSE ORIGINAL single theta MSE cumulative0" } }, { "id": "S016", "uses": [ "G001", "S015" ], "before": "MSE ORIGINAL single theta error row1 := sub(-6, -6)", "after": "MSE ORIGINAL single theta error row1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta error row1，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta error row1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-6", "-6" ], "output": "0", "label": "MSE ORIGINAL single theta error row1" } }, { "id": "S017", "uses": [ "G001", "S016" ], "before": "MSE ORIGINAL single theta squared error row1 := mul(0, 0)", "after": "MSE ORIGINAL single theta squared error row1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta squared error row1，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta squared error row1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "0", "0" ], "output": "0", "label": "MSE ORIGINAL single theta squared error row1" } }, { "id": "S018", "uses": [ "G001", "S017" ], "before": "MSE ORIGINAL single theta weighted error row1 := mul(1/2, 0)", "after": "MSE ORIGINAL single theta weighted error row1 = 0", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta weighted error row1，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta weighted error row1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/2", "0" ], "output": "0", "label": "MSE ORIGINAL single theta weighted error row1" } }, { "id": "S019", "uses": [ "G001", "S018" ], "before": "MSE ORIGINAL single theta MSE cumulative1 := sum(1, 0)", "after": "MSE ORIGINAL single theta MSE cumulative1 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta MSE cumulative1，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta MSE cumulative1 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "0" ], "output": "1", "label": "MSE ORIGINAL single theta MSE cumulative1" } }, { "id": "S020", "uses": [ "G001", "S019" ], "before": "MSE ORIGINAL single theta error row2 := sub(-4, -6)", "after": "MSE ORIGINAL single theta error row2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta error row2，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta error row2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-4", "-6" ], "output": "2", "label": "MSE ORIGINAL single theta error row2" } }, { "id": "S021", "uses": [ "G001", "S020" ], "before": "MSE ORIGINAL single theta squared error row2 := mul(2, 2)", "after": "MSE ORIGINAL single theta squared error row2 = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta squared error row2，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta squared error row2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2", "2" ], "output": "4", "label": "MSE ORIGINAL single theta squared error row2" } }, { "id": "S022", "uses": [ "G001", "S021" ], "before": "MSE ORIGINAL single theta weighted error row2 := mul(1/4, 4)", "after": "MSE ORIGINAL single theta weighted error row2 = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta weighted error row2，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta weighted error row2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "1/4", "4" ], "output": "1", "label": "MSE ORIGINAL single theta weighted error row2" } }, { "id": "S023", "uses": [ "G001", "S022" ], "before": "MSE ORIGINAL single theta MSE cumulative2 := sum(1, 1)", "after": "MSE ORIGINAL single theta MSE cumulative2 = 2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL single theta MSE cumulative2，不以小數近似替代。", "en": "Recompute MSE ORIGINAL single theta MSE cumulative2 with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "1", "1" ], "output": "2", "label": "MSE ORIGINAL single theta MSE cumulative2" } }, { "id": "S024", "uses": [ "G001", "S023" ], "before": "source roles = [MSE ORIGINAL single theta, -6, [[-8, 1/4, 4, 1], [-6, 1/2, 0, 0], [-4, 1/4, 4, 1]]]", "after": "derived = [-6, 2, 否, 否, 是]", "relation": "implies", "rule_id": "MSE_nonnegative_terms_zero_IFF_POSITIVE_mass_equality", "conditions": [], "explanation": { "zh-Hant": "MSE直接加全部加權平方誤差，不用bias-variance分解。各項非負，和0 iff 每個正mass處誤差0；null值可不同。兩邊都False時iff仍True。", "en": "MSE directly sums all weighted squared errors,without a bias-variance decomposition.Nonnegative terms sum to0 iff the error vanishes at every positive-mass value;null values may differ.Both sides False still give a True iff." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S025", "uses": [ "G001", "S024" ], "before": "MSE ORIGINAL observed minus theta := sub(-8, -6)", "after": "MSE ORIGINAL observed minus theta = -2", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL observed minus theta，不以小數近似替代。", "en": "Recompute MSE ORIGINAL observed minus theta with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "-8", "-6" ], "output": "-2", "label": "MSE ORIGINAL observed minus theta" } }, { "id": "S026", "uses": [ "G001", "S025" ], "before": "MSE ORIGINAL one observed squared error := mul(-2, -2)", "after": "MSE ORIGINAL one observed squared error = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。", "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 MSE ORIGINAL one observed squared error，不以小數近似替代。", "en": "Recompute MSE ORIGINAL one observed squared error with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "-2", "-2" ], "output": "4", "label": "MSE ORIGINAL one observed squared error" } }, { "id": "S027", "uses": [ "G001", "S026" ], "before": "source roles = [-6, -8, 4, -6, 2]", "after": "derived = [是, 否, 是, 否]", "relation": "implies", "rule_id": "MSE_observed_loss_unbiasedness_risk_and_consistency_separate", "conditions": [], "explanation": { "zh-Hant": "單一observed loss可偶然等於MSE，也可來自列出的null值；必須仍依全law算risk。E[T]=theta不推出risk0，此處沒有n-indexed漸近law，固定No只不提供consistency證書。", "en": "One observed loss may accidentally equal MSE and may be a listed null value;the risk still uses the whole law.E[T]=theta need not imply zero risk.There is no n-indexed asymptotic law;No only withholds consistency." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ "-6", "-8", "4", "-6", true, "2", false, true, false, "one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency" ] }, "conclusion": { "canonical_value": "(-6, -8, 4, -6, 是, 2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations/questions.jsonl", "batch_sha256": "1e1e01f59905c2360b0733ed5d8983c4bb1e5d37940b1605b3eab38a9bb6e573", "original_line": 82, "original_line_sha256": "fc4bc269631d32d21a07ab92132756cc2791e3db98dafd82d7ab34cf70b21e05", "question_compact_sha256": "d99d58f81fa1ef3efe20d71b9f921f426680debcc553d5e87fbba4b364fa7123", "english_row_compact_sha256": "cb1c978c2ba2751e55c548aeda9d3920cfd6fb3ddc2041bd45147389b9cbbd92", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations.en.jsonl" }, "mko_bindings": { "mko-estimator-mean-squared-error": { "sha256": "b86734a34e0811ecbcceb6973118f1feff5e437465869c8706bc6a00c4cce3cb", "english_sha256": "89d9b8b6f81c362c4f68e6b4cbaf80938cc9168434c135366f56330c231e6f5f", "original_path": "public/data/mko/estimator-mean-squared-error.json", "english_path": "public/data/i18n/editions/mko-estimator-mean-squared-error.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "991c73df066352bfcfb1142a2492e104cd1d20dafe7e9ecea8cff26a4f9d8e50", "original_candidate_sha256": "c468391e8ea3e260f7c5af9f3cfb2f6fe25eac2096430af63c0bc6776ff122ad", "original_candidate_line_sha256": "c468391e8ea3e260f7c5af9f3cfb2f6fe25eac2096430af63c0bc6776ff122ad" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-010-p50", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3", "question_first_witness_sha256": "f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 27, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "b29051a9a6384eb3d9d1c4008e7d35c0c43be01f8403a46ea52f1ef40314eb14", "arithmetic_steps": 23 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }