Question 574: The Observed Likelihood Uses Every Qualified Parameter Law

Original question and givens

Use complete theta labels=[a23, b23], observations=[x23, y23] 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. Observed label=x23. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).

• G001 — Use complete theta labels=[a23, b23], observations=[x23, y23] 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. Observed label=x23. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).

Definition and scope

Each parameter law is normalized. Fix the observation and compare its column; likelihood is not a posterior.

• mko-finite-maximum-likelihood-estimator

Step-by-step derivation

Θ=[a23, b23]; Ω=[x23, y23]

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

P_a23(Ω) := sum(3/4, 1/4)

P_a23(Ω) = 1

P_b23(Ω) := sum(1/4, 3/4)

P_b23(Ω) = 1

parameter=a23; observed=x23

L(a23;x23)=3/4

parameter=b23; observed=x23

L(b23;x23)=1/4

[[a23, 3/4], [b23, 1/4]]

maximum=3/4; all argmax=[a23]

likelihood indexed by fixed observed outcome

not a posterior probability over parameters

Reverse checks

• 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_b23(Ω) with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute P_a23(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

([a23, b23], [x23, y23], [[3/4, 1/4], [1/4, 3/4]], x23, [[a23, 3/4], [b23, 1/4]], complete_normalized_finite_model_likelihood_table)

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.