# Question 163: Retain Every Likelihood Maximizer before Discussing Uniqueness [Return to catalogue page 1](https://ocme.evemisslab.com/en/questions/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations/#q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0038) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. P250-0163 · q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0038 ## Original question and givens Use complete theta labels=[a12, b12], observations=[x12, y12] 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=x12. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate). - G001 — Use complete theta labels=[a12, b12], observations=[x12, y12] 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=x12. 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](https://ocme.evemisslab.com/en/mko/mko-finite-maximum-likelihood-estimator/) ## Step-by-step derivation S001 · impliesBefore Θ=[a12, b12]; Ω=[x12, y12] After observed=x12, column=0 Conditions: none Fix the observed outcome and compare its one probability column across parameter values. Step identifiers and dependenciesRule: fixed_observation · Uses: G001 S002 · calculationBefore P_a12(Ω) := sum(3/4, 1/4) After P_a12(Ω) = 1 Conditions: none First check that this parameter's row remains a normalized probability law. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore row for a12=[3/4, 1/4] After L(a12;x12)=P_a12(x12)=3/4 Conditions: none Take the probability at the observed label in this row, without introducing a prior. Step identifiers and dependenciesRule: likelihood_lookup · Uses: G001, S002 S004 · calculationBefore P_b12(Ω) := sum(1/4, 3/4) After P_b12(Ω) = 1 Conditions: none First check that this parameter's row remains a normalized probability law. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore row for b12=[1/4, 3/4] After L(b12;x12)=P_b12(x12)=1/4 Conditions: none Take the probability at the observed label in this row, without introducing a prior. Step identifiers and dependenciesRule: likelihood_lookup · Uses: G001, S004 S006 · impliesBefore L(a12;x12)=3/4, maximum=3/4 After a12 ∈ argmax: Yes Conditions: none Include every parameter that attains the maximum, including ties after the first maximizer. Step identifiers and dependenciesRule: argmax_membership · Uses: G001, S005 S007 · impliesBefore L(b12;x12)=1/4, maximum=3/4 After b12 ∈ argmax: No Conditions: none Include every parameter that attains the maximum, including ties after the first maximizer. Step identifiers and dependenciesRule: argmax_membership · Uses: G001, S006 S008 · impliesBefore argmax=[a12] After unique maximizing parameter: Yes Conditions: none Only a complete argmax with exactly one element has a unique maximizing parameter. Step identifiers and dependenciesRule: cardinality_one · Uses: G001, S007 S009 · impliesBefore finite normalized laws After MLE is not a posterior or an automatic quality certificate Conditions: none If all likelihoods at an observation are zero, every parameter still maximizes at 0; this computation does not imply unbiasedness or consistency. Step identifiers and dependenciesRule: scope_boundary · Uses: G001, S008 ## 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_b12(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_a12(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 (x12, [[a12, 3/4], [b12, 1/4]], 3/4, [a12], Yes, MLE_argmax_need_not_be_unique) The maximum likelihood is 3/4; the complete argmax is [a12]. ## 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: cc49f05109c3f6ab826d605f01fc57405bceeff2b16ef2dbd9ee84946af65095 - candidate record SHA-256: a98b9615dc539d5a85e39a3fb484b45fe9f4a42d15420f2b04a134468eff2243 - semantic review receipt SHA-256: 4727d1bd957a9b7f6cadb7972ddb5ab80a31f5259dc5394c1dff2b71944ed36b - question-first witness SHA-256: 51e18649133238ac992229f6e16542a5f3af53ca73b2262ab0524508ea937dee - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0038.json)