Question 594: One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator

Original question and givens

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).

• G001 — 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).

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=[a18, b18]; Ω=[x18, y18]

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

P_a18(Ω) := sum(3/4, 1/4)

P_a18(Ω) = 1

P_b18(Ω) := sum(1/4, 3/4)

P_b18(Ω) = 1

tie priority=[a18, b18]

one fixed ordering, independent of the observed outcome

parameter=a18; observed=x18

L(a18;x18)=3/4

parameter=b18; observed=x18

L(b18;x18)=1/4

[[a18, 3/4], [b18, 1/4]]

maximum=3/4; all argmax=[a18]

argmax(x18)=[a18]; priority=[a18, b18]

estimator(x18)=a18

parameter=a18; observed=y18

L(a18;y18)=1/4

parameter=b18; observed=y18

L(b18;y18)=3/4

[[a18, 1/4], [b18, 3/4]]

maximum=3/4; all argmax=[b18]

argmax(y18)=[b18]; priority=[a18, b18]

estimator(y18)=b18

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_b18(Ω) with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute P_a18(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

([a18, b18], [[x18, [a18], a18], [y18, [b18], b18]], 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.