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Independent leaf mixtures preserve actual pair-event masses

Lax342547.IndependentMixtures · concepts/Lax342547/IndependentMixtures.lean · lax-342547

proven

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    Natural Language Statement

    Lemma

    The product of two actual finite leaf mixtures is exactly the mixture of their independent leaf-pair laws, weighted by the original mass product. The same identity holds for every pair event.

    Concept map
    9 concepts; 1 descendant hidden
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 independent_mixture_event proven

    2 independent_pair_mixture proven

    Lean source view on GitHub

    1import Lax342547.PairRecovery
    2import Lax342547.MixtureRecovery
    3
    4/-!
    5---
    6title: Independent leaf mixtures preserve actual pair-event masses
    7type: lemma
    8---
    9The product of two actual finite leaf mixtures is exactly the mixture of their independent leaf-pair laws, weighted by the original mass product. The same identity holds for every pair event.
    10-/
    11
    12namespace Lax342547.IndependentMixtures
    13
    14open Lax342547.PairRecovery
    15open scoped ENNReal
    16
    17axiom independent_pair_mixture {ΩA ΩB J K : Type}
    18 [Fintype ΩA] [Fintype ΩB] [Fintype J] [Fintype K]
    19 (p : J → PMF ΩA) (q : K → PMF ΩB) (p₀ : PMF ΩA) (q₀ : PMF ΩB)
    20 (w : J → ℝ≥0∞) (v : K → ℝ≥0∞)
    21 (hp : ∀ a, p₀ a = ∑ j, w j * p j a) (hq : ∀ b, q₀ b = ∑ k, v k * q k b)
    22 (ab : ΩA × ΩB) :
    23 independentPair p₀ q₀ ab = ∑ jk : J × K, (w jk.1 * v jk.2) * independentPair (p jk.1) (q jk.2) ab
    24
    25axiom independent_mixture_event {ΩA ΩB J K : Type}
    26 [Fintype ΩA] [Fintype ΩB] [Fintype J] [Fintype K]
    27 (p : J → PMF ΩA) (q : K → PMF ΩB) (p₀ : PMF ΩA) (q₀ : PMF ΩB)
    28 (w : J → ℝ≥0∞) (v : K → ℝ≥0∞)
    29 (hp : ∀ a, p₀ a = ∑ j, w j * p j a) (hq : ∀ b, q₀ b = ∑ k, v k * q k b)
    30 (H : Set (ΩA × ΩB)) :
    31 (independentPair p₀ q₀).toOuterMeasure H =
    32 ∑ jk : J × K, (w jk.1 * v jk.2) * (independentPair (p jk.1) (q jk.2)).toOuterMeasure H
    33
    34end Lax342547.IndependentMixtures
    35
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