Question 597: One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator

Original question and givens

Use complete theta labels=[a21, b21], observations=[x21, y21] 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=[a21, b21] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).

• G001 — Use complete theta labels=[a21, b21], observations=[x21, y21] 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=[a21, b21] 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=[a21, b21]; Ω=[x21, y21]

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

P_a21(Ω) := sum(3/4, 1/4)

P_a21(Ω) = 1

P_b21(Ω) := sum(1/4, 3/4)

P_b21(Ω) = 1

tie priority=[a21, b21]

one fixed ordering, independent of the observed outcome

parameter=a21; observed=x21

L(a21;x21)=3/4

parameter=b21; observed=x21

L(b21;x21)=1/4

[[a21, 3/4], [b21, 1/4]]

maximum=3/4; all argmax=[a21]

argmax(x21)=[a21]; priority=[a21, b21]

estimator(x21)=a21

parameter=a21; observed=y21

L(a21;y21)=1/4

parameter=b21; observed=y21

L(b21;y21)=3/4

[[a21, 1/4], [b21, 3/4]]

maximum=3/4; all argmax=[b21]

argmax(y21)=[b21]; priority=[a21, b21]

estimator(y21)=b21

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_b21(Ω) with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute P_a21(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

([a21, b21], [[x21, [a21], a21], [y21, [b21], b21]], 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.