Question 146: Conditional Laws Are Defined Only on Positive-Mass Statistic Cells

Original question and givens

Use complete theta labels=[a20, b20], samples=[200, 201, 202, 203], 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=[a20, b20], samples=[200, 201, 202, 203], 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

Step-by-step derivation

Ω=[200, 201, 202, 203]; T=[0, 1, 0, 1]

fibers=[[0, [200, 202]], [1, [201, 203]]]

[[0, [200, 202]], [1, [201, 203]]]

disjoint union of fibers = [200, 201, 202, 203]

P_a20(Ω) := sum(2/5, 1/10, 2/5, 1/10)

P_a20(Ω) = 1

P_a20(T=0) := sum(2/5, 2/5)

P_a20(T=0) = 4/5

P_a20(X=200|T=0) := div(2/5, 4/5)

P_a20(X=200|T=0) = 1/2

reconstruct P_a20(X=200) := mul(1/2, 4/5)

reconstruct P_a20(X=200) = 2/5

P_a20(X=202|T=0) := div(2/5, 4/5)

P_a20(X=202|T=0) = 1/2

reconstruct P_a20(X=202) := mul(1/2, 4/5)

reconstruct P_a20(X=202) = 2/5

conditional total (a20,0) := sum(1/2, 1/2)

conditional total (a20,0) = 1

P_a20(T=1) := sum(1/10, 1/10)

P_a20(T=1) = 1/5

P_a20(X=201|T=1) := div(1/10, 1/5)

P_a20(X=201|T=1) = 1/2

reconstruct P_a20(X=201) := mul(1/2, 1/5)

reconstruct P_a20(X=201) = 1/10

P_a20(X=203|T=1) := div(1/10, 1/5)

P_a20(X=203|T=1) = 1/2

reconstruct P_a20(X=203) := mul(1/2, 1/5)

reconstruct P_a20(X=203) = 1/10

conditional total (a20,1) := sum(1/2, 1/2)

conditional total (a20,1) = 1

P_b20(Ω) := sum(1/10, 2/5, 1/10, 2/5)

P_b20(Ω) = 1

P_b20(T=0) := sum(1/10, 1/10)

P_b20(T=0) = 1/5

P_b20(X=200|T=0) := div(1/10, 1/5)

P_b20(X=200|T=0) = 1/2

reconstruct P_b20(X=200) := mul(1/2, 1/5)

reconstruct P_b20(X=200) = 1/10

P_b20(X=202|T=0) := div(1/10, 1/5)

P_b20(X=202|T=0) = 1/2

reconstruct P_b20(X=202) := mul(1/2, 1/5)

reconstruct P_b20(X=202) = 1/10

conditional total (b20,0) := sum(1/2, 1/2)

conditional total (b20,0) = 1

P_b20(T=1) := sum(2/5, 2/5)

P_b20(T=1) = 4/5

P_b20(X=201|T=1) := div(2/5, 4/5)

P_b20(X=201|T=1) = 1/2

reconstruct P_b20(X=201) := mul(1/2, 4/5)

reconstruct P_b20(X=201) = 2/5

P_b20(X=203|T=1) := div(2/5, 4/5)

P_b20(X=203|T=1) = 1/2

reconstruct P_b20(X=203) := mul(1/2, 4/5)

reconstruct P_b20(X=203) = 2/5

conditional total (b20,1) := sum(1/2, 1/2)

conditional total (b20,1) = 1

all parameter × fiber cells audited

[[a20, 0, 4/5, [[200, 1/2], [202, 1/2]]], [a20, 1, 1/5, [[201, 1/2], [203, 1/2]]], [b20, 0, 1/5, [[200, 1/2], [202, 1/2]]], [b20, 1, 4/5, [[201, 1/2], [203, 1/2]]]]

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 (b20,1) with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute reconstruct P_b20(X=203) with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute P_b20(X=203|T=1) with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute reconstruct P_b20(X=201) with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute P_b20(X=201|T=1) with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute P_b20(T=1) with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute conditional total (b20,0) with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute reconstruct P_b20(X=202) with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute P_b20(X=202|T=0) with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute reconstruct P_b20(X=200) with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute P_b20(X=200|T=0) with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute P_b20(T=0) with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute P_b20(Ω) with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute conditional total (a20,1) with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute reconstruct P_a20(X=203) with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute P_a20(X=203|T=1) with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute reconstruct P_a20(X=201) with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute P_a20(X=201|T=1) with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute P_a20(T=1) with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute conditional total (a20,0) with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute reconstruct P_a20(X=202) with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute P_a20(X=202|T=0) with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute reconstruct P_a20(X=200) with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute P_a20(X=200|T=0) with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute P_a20(T=0) with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute P_a20(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

([[a20, 0, 4/5, [[200, 1/2], [202, 1/2]]], [a20, 1, 1/5, [[201, 1/2], [203, 1/2]]], [b20, 0, 1/5, [[200, 1/2], [202, 1/2]]], [b20, 1, 4/5, [[201, 1/2], [203, 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.