Complete Likelihood Argmax Sets and Fixed Total Tie Rules
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
2. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
3. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
4. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
5. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
6. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
7. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
8. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
9. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
10. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
11. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
12. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
13. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
14. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
15. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
16. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
17. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
18. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
19. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
20. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
21. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
22. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
23. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
24. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
25. The Observed Likelihood Uses Every Qualified Parameter Law
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-law-likelihood · candidate
26. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x0, [[a0, 3/4], [b0, 1/4]], 3/4, [a0], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
27. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x1, [[a1, 3/4], [b1, 1/4]], 3/4, [a1], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
28. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x2, [[a2, 3/4], [b2, 1/4]], 3/4, [a2], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
29. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x3, [[a3, 3/4], [b3, 1/4]], 3/4, [a3], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
30. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x4, [[a4, 3/4], [b4, 1/4]], 3/4, [a4], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
31. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x5, [[a5, 3/4], [b5, 1/4]], 3/4, [a5], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
32. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x6, [[a6, 3/4], [b6, 1/4]], 3/4, [a6], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
33. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x7, [[a7, 3/4], [b7, 1/4]], 3/4, [a7], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
34. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x8, [[a8, 3/4], [b8, 1/4]], 3/4, [a8], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
35. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x9, [[a9, 3/4], [b9, 1/4]], 3/4, [a9], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
36. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x10, [[a10, 3/4], [b10, 1/4]], 3/4, [a10], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
37. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x11, [[a11, 3/4], [b11, 1/4]], 3/4, [a11], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
38. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x12, [[a12, 3/4], [b12, 1/4]], 3/4, [a12], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
39. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x13, [[a13, 3/4], [b13, 1/4]], 3/4, [a13], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
40. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x14, [[a14, 3/4], [b14, 1/4]], 3/4, [a14], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
41. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x15, [[a15, 3/4], [b15, 1/4]], 3/4, [a15], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
42. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x16, [[a16, 3/4], [b16, 1/4]], 3/4, [a16], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
43. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x17, [[a17, 3/4], [b17, 1/4]], 3/4, [a17], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
44. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x18, [[a18, 3/4], [b18, 1/4]], 3/4, [a18], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
45. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x19, [[a19, 3/4], [b19, 1/4]], 3/4, [a19], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
46. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x20, [[a20, 3/4], [b20, 1/4]], 3/4, [a20], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
47. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x21, [[a21, 3/4], [b21, 1/4]], 3/4, [a21], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
48. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x22, [[a22, 3/4], [b22, 1/4]], 3/4, [a22], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
49. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x23, [[a23, 3/4], [b23, 1/4]], 3/4, [a23], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
50. Retain Every Likelihood Maximizer before Discussing Uniqueness
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).
View answer and explanation
(x24, [[a24, 3/4], [b24, 1/4]], 3/4, [a24], Yes, MLE_argmax_need_not_be_unique)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-argmax · candidate
51. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a0, b0], [[x0, [a0], a0], [y0, [b0], b0]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
52. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a1, b1], [[x1, [a1], a1], [y1, [b1], b1]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
53. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a2, b2], [[x2, [a2], a2], [y2, [b2], b2]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
54. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a3, b3], [[x3, [a3], a3], [y3, [b3], b3]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
55. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a4, b4], [[x4, [a4], a4], [y4, [b4], b4]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
56. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a5, b5], [[x5, [a5], a5], [y5, [b5], b5]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
57. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a6, b6], [[x6, [a6], a6], [y6, [b6], b6]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
58. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a7, b7], [[x7, [a7], a7], [y7, [b7], b7]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
59. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a8, b8], [[x8, [a8], a8], [y8, [b8], b8]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
60. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a9, b9], [[x9, [a9], a9], [y9, [b9], b9]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
61. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a10, b10], [[x10, [a10], a10], [y10, [b10], b10]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
62. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a11, b11], [[x11, [a11], a11], [y11, [b11], b11]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
63. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a12, b12], [[x12, [a12], a12], [y12, [b12], b12]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
64. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a13, b13], [[x13, [a13], a13], [y13, [b13], b13]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
65. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a14, b14], [[x14, [a14], a14], [y14, [b14], b14]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
66. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a15, b15], [[x15, [a15], a15], [y15, [b15], b15]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
67. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a16, b16], [[x16, [a16], a16], [y16, [b16], b16]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
68. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a17, b17], [[x17, [a17], a17], [y17, [b17], b17]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
69. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a18, b18], [[x18, [a18], a18], [y18, [b18], b18]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
70. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a19, b19], [[x19, [a19], a19], [y19, [b19], b19]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
71. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a20, b20], [[x20, [a20], a20], [y20, [b20], b20]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
72. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a21, b21], [[x21, [a21], a21], [y21, [b21], b21]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
73. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a22, b22], [[x22, [a22], a22], [y22, [b22], b22]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
74. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a23, b23], [[x23, [a23], a23], [y23, [b23], b23]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
75. One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator
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).
View answer and explanation
([a24, b24], [[x24, [a24], a24], [y24, [b24], b24]], Yes, theta_independent_tie_rule_makes_a_total_estimator)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-total-tie-estimator · candidate
76. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
77. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
78. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
79. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
80. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
81. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
82. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
83. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
84. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
85. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
86. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
87. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
88. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
89. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
90. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
91. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
92. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
93. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
94. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
95. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
96. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
97. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
98. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
99. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
100. Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule
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).
When every theta has zero likelihood, every theta is still an argmax and the fixed tie rule selects a total value. No posterior or estimator-quality conclusion is invented.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Maximum-likelihood estimator in a finite model · qf-mod47-mle-boundary · candidate
Canonical question JSONL · English JSONL · Validation report