Question 567: The Observed Likelihood Uses Every Qualified Parameter Law

Original question and givens

Use complete theta labels=[a16, b16], observations=[x16, y16] 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=x16. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate).

• G001 — Use complete theta labels=[a16, b16], observations=[x16, y16] 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=x16. 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=[a16, b16]; Ω=[x16, y16]

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

P_a16(Ω) := sum(3/4, 1/4)

P_a16(Ω) = 1

P_b16(Ω) := sum(1/4, 3/4)

P_b16(Ω) = 1

parameter=a16; observed=x16

L(a16;x16)=3/4

parameter=b16; observed=x16

L(b16;x16)=1/4

[[a16, 3/4], [b16, 1/4]]

maximum=3/4; all argmax=[a16]

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_b16(Ω) with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute P_a16(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

([a16, b16], [x16, y16], [[3/4, 1/4], [1/4, 3/4]], x16, [[a16, 3/4], [b16, 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.