Question 152: Retain Every Likelihood Maximizer before Discussing Uniqueness

Original question and givens

Use complete theta labels=[a1, b1], observations=[x1, y1] 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=x1. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).

• G001 — Use complete theta labels=[a1, b1], observations=[x1, y1] 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=x1. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).

Definition and scope

Use complete finite parameter and observation carriers. Fix the observed column, retain the full argmax, and introduce no prior.

• mko-finite-maximum-likelihood-estimator

Step-by-step derivation

Θ=[a1, b1]; Ω=[x1, y1]

observed=x1, column=0

P_a1(Ω) := sum(3/4, 1/4)

P_a1(Ω) = 1

row for a1=[3/4, 1/4]

L(a1;x1)=P_a1(x1)=3/4

P_b1(Ω) := sum(1/4, 3/4)

P_b1(Ω) = 1

row for b1=[1/4, 3/4]

L(b1;x1)=P_b1(x1)=1/4

L(a1;x1)=3/4, maximum=3/4

a1 ∈ argmax: Yes

L(b1;x1)=1/4, maximum=3/4

b1 ∈ argmax: No

argmax=[a1]

unique maximizing parameter: Yes

finite normalized laws

MLE is not a posterior or an automatic quality certificate

Reverse checks

• S009 · premise_and_result_audit — Do not confuse a null observation column with an invalid all-zero probability-law row.
• S008 · premise_and_result_audit — A fixed tie-breaker can select one output without making a multi-element argmax a singleton.
• S007 · premise_and_result_audit — The maximum is a number; argmax is a set of parameters.
• S006 · premise_and_result_audit — The maximum is a number; argmax is a set of parameters.
• S005 · premise_and_result_audit — The value remains bound to the same parameter and observation.
• S004 · premise_and_result_audit — Recompute P_b1(Ω) with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — The value remains bound to the same parameter and observation.
• S002 · premise_and_result_audit — Recompute P_a1(Ω) with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Do not confuse other observations in a row with other parameter values.

Result

(x1, [[a1, 3/4], [b1, 1/4]], 3/4, [a1], Yes, MLE_argmax_need_not_be_unique)

The maximum likelihood is 3/4; the complete argmax is [a1].

This question review and source binding

Review date: 2026-10-24

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.