# Question 126: Conditional Laws Are Defined Only on Positive-Mass Statistic Cells [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-r05-finite-sufficient-statistic-foundations/#q-20261015-advance-20261116-r05-finite-sufficient-statistic-0026) 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. P250-0126 · q-20261015-advance-20261116-r05-finite-sufficient-statistic-0026 ## Original question and givens Use complete theta labels=[a0, b0], samples=[0, 1, 2, 3], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary). - G001 — Use complete theta labels=[a0, b0], samples=[0, 1, 2, 3], aligned exact probability rows=[[[2, 5], [1, 10], [2, 5], [1, 10]], [[1, 10], [2, 5], [1, 10], [2, 5]]] and total statistic=[0, 1, 0, 1]. Each theta law is normalized. Fibers partition the full sample carrier, retaining zero-mass cells. Give (conditional_cells, boundary). ## Definition and scope The statistic is total on the full sample carrier. Normalize each parameter law and condition only on positive-mass fibers. - [mko-finite-sufficient-statistic](https://ocme.evemisslab.com/en/mko/mko-finite-sufficient-statistic/) ## Step-by-step derivation S001 · impliesBefore Ω=[0, 1, 2, 3]; T=[0, 1, 0, 1] After fibers=[[0, [0, 2]], [1, [1, 3]]] Conditions: none Group the original sample labels by statistic value; distinguish a sample's position from its label. Step identifiers and dependenciesRule: statistic_fibers · Uses: G001 S002 · impliesBefore [[0, [0, 2]], [1, [1, 3]]] After disjoint union of fibers = [0, 1, 2, 3] Conditions: none Each sample lies in exactly one fiber, and their union covers all of Omega. Step identifiers and dependenciesRule: partition_audit · Uses: G001, S001 S003 · calculationBefore P_a0(Ω) := sum(2/5, 1/10, 2/5, 1/10) After P_a0(Ω) = 1 Conditions: none Verify normalization of the probability row for this parameter. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore P_a0(T=0) := sum(2/5, 2/5) After P_a0(T=0) = 4/5 Conditions: none Sum only the masses in fiber [0, 2]. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore P_a0(X=0|T=0) := div(2/5, 4/5) After P_a0(X=0|T=0) = 1/2 Conditions: none The numerator is the atom mass 2/5 under the same law and the denominator is the fiber mass 4/5. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore reconstruct P_a0(X=0) := mul(1/2, 4/5) After reconstruct P_a0(X=0) = 2/5 Conditions: none Multiply by the fiber mass to recover that atom's original probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore P_a0(X=2|T=0) := div(2/5, 4/5) After P_a0(X=2|T=0) = 1/2 Conditions: none The numerator is the atom mass 2/5 under the same law and the denominator is the fiber mass 4/5. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore reconstruct P_a0(X=2) := mul(1/2, 4/5) After reconstruct P_a0(X=2) = 2/5 Conditions: none Multiply by the fiber mass to recover that atom's original probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore conditional total (a0,0) := sum(1/2, 1/2) After conditional total (a0,0) = 1 Conditions: none The full conditional law on a positive-mass fiber must sum to 1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore P_a0(T=1) := sum(1/10, 1/10) After P_a0(T=1) = 1/5 Conditions: none Sum only the masses in fiber [1, 3]. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore P_a0(X=1|T=1) := div(1/10, 1/5) After P_a0(X=1|T=1) = 1/2 Conditions: none The numerator is the atom mass 1/10 under the same law and the denominator is the fiber mass 1/5. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore reconstruct P_a0(X=1) := mul(1/2, 1/5) After reconstruct P_a0(X=1) = 1/10 Conditions: none Multiply by the fiber mass to recover that atom's original probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore P_a0(X=3|T=1) := div(1/10, 1/5) After P_a0(X=3|T=1) = 1/2 Conditions: none The numerator is the atom mass 1/10 under the same law and the denominator is the fiber mass 1/5. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore reconstruct P_a0(X=3) := mul(1/2, 1/5) After reconstruct P_a0(X=3) = 1/10 Conditions: none Multiply by the fiber mass to recover that atom's original probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore conditional total (a0,1) := sum(1/2, 1/2) After conditional total (a0,1) = 1 Conditions: none The full conditional law on a positive-mass fiber must sum to 1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore P_b0(Ω) := sum(1/10, 2/5, 1/10, 2/5) After P_b0(Ω) = 1 Conditions: none Verify normalization of the probability row for this parameter. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore P_b0(T=0) := sum(1/10, 1/10) After P_b0(T=0) = 1/5 Conditions: none Sum only the masses in fiber [0, 2]. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore P_b0(X=0|T=0) := div(1/10, 1/5) After P_b0(X=0|T=0) = 1/2 Conditions: none The numerator is the atom mass 1/10 under the same law and the denominator is the fiber mass 1/5. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore reconstruct P_b0(X=0) := mul(1/2, 1/5) After reconstruct P_b0(X=0) = 1/10 Conditions: none Multiply by the fiber mass to recover that atom's original probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore P_b0(X=2|T=0) := div(1/10, 1/5) After P_b0(X=2|T=0) = 1/2 Conditions: none The numerator is the atom mass 1/10 under the same law and the denominator is the fiber mass 1/5. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore reconstruct P_b0(X=2) := mul(1/2, 1/5) After reconstruct P_b0(X=2) = 1/10 Conditions: none Multiply by the fiber mass to recover that atom's original probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore conditional total (b0,0) := sum(1/2, 1/2) After conditional total (b0,0) = 1 Conditions: none The full conditional law on a positive-mass fiber must sum to 1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore P_b0(T=1) := sum(2/5, 2/5) After P_b0(T=1) = 4/5 Conditions: none Sum only the masses in fiber [1, 3]. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore P_b0(X=1|T=1) := div(2/5, 4/5) After P_b0(X=1|T=1) = 1/2 Conditions: none The numerator is the atom mass 2/5 under the same law and the denominator is the fiber mass 4/5. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore reconstruct P_b0(X=1) := mul(1/2, 4/5) After reconstruct P_b0(X=1) = 2/5 Conditions: none Multiply by the fiber mass to recover that atom's original probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore P_b0(X=3|T=1) := div(2/5, 4/5) After P_b0(X=3|T=1) = 1/2 Conditions: none The numerator is the atom mass 2/5 under the same law and the denominator is the fiber mass 4/5. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore reconstruct P_b0(X=3) := mul(1/2, 4/5) After reconstruct P_b0(X=3) = 2/5 Conditions: none Multiply by the fiber mass to recover that atom's original probability. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore conditional total (b0,1) := sum(1/2, 1/2) After conditional total (b0,1) = 1 Conditions: none The full conditional law on a positive-mass fiber must sum to 1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · impliesBefore all parameter × fiber cells audited After [[a0, 0, 4/5, [[0, 1/2], [2, 1/2]]], [a0, 1, 1/5, [[1, 1/2], [3, 1/2]]], [b0, 0, 1/5, [[0, 1/2], [2, 1/2]]], [b0, 1, 4/5, [[1, 1/2], [3, 1/2]]]] Conditions: none Return every cell in parameter order and then statistic-value order; retain the zero-mass branch rule. Step identifiers and dependenciesRule: complete_conditional_table · Uses: G001, S028 ## Reverse checks - S029 · premise_and_result_audit — This task computes the conditional table; it makes no estimator-quality claim and does not invoke factorization. - S028 · premise_and_result_audit — Recompute conditional total (b0,1) with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute reconstruct P_b0(X=3) with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute P_b0(X=3|T=1) with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute reconstruct P_b0(X=1) with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute P_b0(X=1|T=1) with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute P_b0(T=1) with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute conditional total (b0,0) with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute reconstruct P_b0(X=2) with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute P_b0(X=2|T=0) with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute reconstruct P_b0(X=0) with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute P_b0(X=0|T=0) with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute P_b0(T=0) with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute P_b0(Ω) with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute conditional total (a0,1) with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute reconstruct P_a0(X=3) with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute P_a0(X=3|T=1) with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute reconstruct P_a0(X=1) with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute P_a0(X=1|T=1) with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute P_a0(T=1) with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute conditional total (a0,0) with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute reconstruct P_a0(X=2) with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute P_a0(X=2|T=0) with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute reconstruct P_a0(X=0) with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute P_a0(X=0|T=0) with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute P_a0(T=0) with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute P_a0(Ω) with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — A fiber stays in the partition even when it has zero mass under a parameter. - S001 · premise_and_result_audit — T must have exactly one value at every sample point. ## Result ([[a0, 0, 4/5, [[0, 1/2], [2, 1/2]]], [a0, 1, 1/5, [[1, 1/2], [3, 1/2]]], [b0, 0, 1/5, [[0, 1/2], [2, 1/2]]], [b0, 1, 4/5, [[1, 1/2], [3, 1/2]]]], zero_mass_cells_remain_undefined) The full cellwise conditional laws are listed; zero-mass cells must retain null. ## 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: ddbda5008ebf2a659e44b59f3cad93074629ef7c7840241418b3849478f1d1ed - candidate record SHA-256: dc1022c91ba3fc83388e3b3691ed6cd196df77ee93dcccbc4c2c1832efed1c23 - 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-r05-finite-sufficient-statistic-0026.json)