Question 618: Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule

Original question and givens

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).

• G001 — 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).

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=[a17, b17]; Ω=[x17, y17]

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

P_a17(Ω) := sum(1/2, 1/2)

P_a17(Ω) = 1

P_b17(Ω) := sum(1/2, 1/2)

P_b17(Ω) = 1

tie priority=[a17, b17]

one fixed ordering, independent of the observed outcome

parameter=a17; observed=x17

L(a17;x17)=1/2

parameter=b17; observed=x17

L(b17;x17)=1/2

[[a17, 1/2], [b17, 1/2]]

maximum=1/2; all argmax=[a17, b17]

argmax(x17)=[a17, b17]; priority=[a17, b17]

estimator(x17)=a17

parameter=a17; observed=y17

L(a17;y17)=1/2

parameter=b17; observed=y17

L(b17;y17)=1/2

[[a17, 1/2], [b17, 1/2]]

maximum=1/2; all argmax=[a17, b17]

argmax(y17)=[a17, b17]; priority=[a17, b17]

estimator(y17)=a17

defined on 2 outcomes

total on all 2 listed outcomes

boundary mode=tie; argmax=[a17, b17]

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_b17(Ω) with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute P_a17(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, 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)

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.