Question 589: One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator

Original question and givens

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).

• G001 — 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).

Definition and scope

Keep each observation's complete argmax, then pick the first member in one fixed parameter-independent tie order, giving a total estimator.

• mko-finite-maximum-likelihood-estimator

Step-by-step derivation

Θ=[a13, b13]; Ω=[x13, y13]

[[3/4, 1/4], [1/4, 3/4]]

P_a13(Ω) := sum(3/4, 1/4)

P_a13(Ω) = 1

P_b13(Ω) := sum(1/4, 3/4)

P_b13(Ω) = 1

tie priority=[a13, b13]

one fixed ordering, independent of the observed outcome

parameter=a13; observed=x13

L(a13;x13)=3/4

parameter=b13; observed=x13

L(b13;x13)=1/4

[[a13, 3/4], [b13, 1/4]]

maximum=3/4; all argmax=[a13]

argmax(x13)=[a13]; priority=[a13, b13]

estimator(x13)=a13

parameter=a13; observed=y13

L(a13;y13)=1/4

parameter=b13; observed=y13

L(b13;y13)=3/4

[[a13, 1/4], [b13, 3/4]]

maximum=3/4; all argmax=[b13]

argmax(y13)=[b13]; priority=[a13, b13]

estimator(y13)=b13

defined on 2 outcomes

total on all 2 listed outcomes

Reverse checks

• 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_b13(Ω) with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute P_a13(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

([a13, b13], [[x13, [a13], a13], [y13, [b13], b13]], Yes, theta_independent_tie_rule_makes_a_total_estimator)

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.