Question 610: Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule

Original question and givens

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).

• G001 — 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).

Definition and scope

Ties and all-zero likelihoods still have a complete argmax. The final No declines posterior or estimator-quality claims.

• mko-finite-maximum-likelihood-estimator

Step-by-step derivation

Θ=[a9, b9]; Ω=[x9, y9]

[[1/2, 1/2], [1/2, 1/2]]

P_a9(Ω) := sum(1/2, 1/2)

P_a9(Ω) = 1

P_b9(Ω) := sum(1/2, 1/2)

P_b9(Ω) = 1

tie priority=[a9, b9]

one fixed ordering, independent of the observed outcome

parameter=a9; observed=x9

L(a9;x9)=1/2

parameter=b9; observed=x9

L(b9;x9)=1/2

[[a9, 1/2], [b9, 1/2]]

maximum=1/2; all argmax=[a9, b9]

argmax(x9)=[a9, b9]; priority=[a9, b9]

estimator(x9)=a9

parameter=a9; observed=y9

L(a9;y9)=1/2

parameter=b9; observed=y9

L(b9;y9)=1/2

[[a9, 1/2], [b9, 1/2]]

maximum=1/2; all argmax=[a9, b9]

argmax(y9)=[a9, b9]; priority=[a9, b9]

estimator(y9)=a9

defined on 2 outcomes

total on all 2 listed outcomes

boundary mode=tie; argmax=[a9, b9]

no automatic posterior, unbiasedness, consistency or asymptotic optimum

Reverse checks

• S014 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S013 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S012 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S011 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S010 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S009 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S008 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S007 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S006 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S005 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S004 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
• S003 · premise_and_result_audit — Recompute P_b9(Ω) with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute P_a9(Ω) with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.

Result

(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)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.