# Question 562: The Observed Likelihood Uses Every Qualified Parameter Law [Return to catalogue page 3](https://ocme.evemisslab.com/en/questions/page/3/) [← 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-0012) 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. P1000-0562 · q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0012 ## Original question and givens Use complete theta labels=[a11, b11], observations=[x11, y11] 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=x11. Give (theta_values, observations, probability_rows, observed, likelihood_rows, certificate). - G001 — Use complete theta labels=[a11, b11], observations=[x11, y11] 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=x11. 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](https://ocme.evemisslab.com/en/mko/mko-finite-maximum-likelihood-estimator/) ## Step-by-step derivation S001 · impliesBefore Θ=[a11, b11]; Ω=[x11, y11] After [[3/4, 1/4], [1/4, 3/4]] Conditions: none Each row is one parameter law and each column an outcome; reorder probabilities together with labels. Step identifiers and dependenciesRule: full_parametric_model · Uses: G001 S002 · calculationBefore P_a11(Ω) := sum(3/4, 1/4) After P_a11(Ω) = 1 Conditions: none Check this law's complete total mass first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore P_b11(Ω) := sum(1/4, 3/4) After P_b11(Ω) = 1 Conditions: none Check this law's complete total mass first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore parameter=a11; observed=x11 After L(a11;x11)=3/4 Conditions: none Fix the observation and compare its column across parameters, without introducing a prior. Step identifiers and dependenciesRule: fixed_likelihood_lookup · Uses: G001, S003 S005 · impliesBefore parameter=b11; observed=x11 After L(b11;x11)=1/4 Conditions: none Fix the observation and compare its column across parameters, without introducing a prior. Step identifiers and dependenciesRule: fixed_likelihood_lookup · Uses: G001, S004 S006 · impliesBefore [[a11, 3/4], [b11, 1/4]] After maximum=3/4; all argmax=[a11] Conditions: none Retain every maximizer; for an all-zero column the complete argmax is still all parameters. Step identifiers and dependenciesRule: complete_argmax · Uses: G001, S005 S007 · impliesBefore likelihood indexed by fixed observed outcome After not a posterior probability over parameters Conditions: none Likelihoods across parameters need not sum to1; the original laws are the normalized rows. Step identifiers and dependenciesRule: likelihood_scope · Uses: G001, S006 ## 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_b11(Ω) with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute P_a11(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 ([a11, b11], [x11, y11], [[3/4, 1/4], [1/4, 3/4]], x11, [[a11, 3/4], [b11, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 647e77292e5fa3ada59fa4a230fafdc030fdf4ca3967e6659c3664fd6ad2f7ae - candidate record SHA-256: 0047f359cd93c6dd779ea8c77b4b81b98831fdfb6f882a632ee473c5859d2617 - semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772 - question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b - 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-0012.json)