Question 601: Tied and All-Zero Likelihoods Still Need a Fixed Total Tie Rule

Original question and givens

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).

• G001 — 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).

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=[a0, b0]; Ω=[x0, y0]

[[0, 1], [0, 1]]

P_a0(Ω) := sum(0, 1)

P_a0(Ω) = 1

P_b0(Ω) := sum(0, 1)

P_b0(Ω) = 1

tie priority=[a0, b0]

one fixed ordering, independent of the observed outcome

parameter=a0; observed=x0

L(a0;x0)=0

parameter=b0; observed=x0

L(b0;x0)=0

[[a0, 0], [b0, 0]]

maximum=0; all argmax=[a0, b0]

argmax(x0)=[a0, b0]; priority=[a0, b0]

estimator(x0)=a0

parameter=a0; observed=y0

L(a0;y0)=1

parameter=b0; observed=y0

L(b0;y0)=1

[[a0, 1], [b0, 1]]

maximum=1; all argmax=[a0, b0]

argmax(y0)=[a0, b0]; priority=[a0, b0]

estimator(y0)=a0

defined on 2 outcomes

total on all 2 listed outcomes

boundary mode=all_zero_likelihood; argmax=[a0, b0]

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_b0(Ω) with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute P_a0(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

(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)

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.