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Actual-mass recovery and residual bounds through finite splits

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

proven

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    Natural Language Statement

    Theorem

    Finite mixture event masses retain their original weights. Successive conditional parts recover the combined restriction exactly. Relative residual budgets on split fibers sum to the same global budget.

    Concept map
    2 concepts; 2 descendants hidden
    100%
    Proven claimThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 conditional_event_mass proven

    2 fiber_residual proven

    3 mixture_event proven

    4 nested_weighted_filter proven

    Lean source view on GitHub

    1import Lax342547.Conditioning
    2
    3/-!
    4---
    5title: Actual-mass recovery and residual bounds through finite splits
    6type: theorem
    7---
    8Finite mixture event masses retain their original weights. Successive
    9conditional parts recover the combined restriction exactly. Relative
    10residual budgets on split fibers sum to the same global budget.
    11-/
    12
    13namespace Lax342547.MixtureRecovery
    14
    15open scoped ENNReal
    16
    17axiom mixture_event {Ω J : Type} [Fintype Ω] [Fintype J]
    18 (p : J → PMF Ω) (q : PMF Ω) (w : J → ℝ≥0∞)
    19 (hq : ∀ o, q o = ∑ j, w j * p j o) (S : Set Ω) :
    20 q.toOuterMeasure S = ∑ j, w j * (p j).toOuterMeasure S
    21
    22axiom nested_weighted_filter {Ω : Type} (p : PMF Ω) (S T : Set Ω)
    23 (hS : ∃ o ∈ S, o ∈ p.support)
    24 (hT : ∃ o ∈ T, o ∈ (p.filter S hS).support) (o : Ω) :
    25 p.toOuterMeasure S * (p.filter S hS).toOuterMeasure T * ((p.filter S hS).filter T hT) o =
    26 (S ∩ T).indicator p o
    27
    28axiom conditional_event_mass {Ω : Type} [Fintype Ω] (p : PMF Ω)
    29 (S T : Set Ω) (hS : ∃ o ∈ S, o ∈ p.support) (hT : T ⊆ S)
    30 (δ : ℝ≥0∞) (hb : (p.filter S hS).toOuterMeasure T ≤ δ) :
    31 p.toOuterMeasure T ≤ δ * p.toOuterMeasure S
    32
    33axiom fiber_residual {Ω K : Type} [Fintype K] (p : PMF Ω) (f : Ω → K)
    34 (R : K → Set Ω) (τ δ : ℝ≥0∞)
    35 (hr : ∀ k, τ ≤ p.toOuterMeasure {o | f o = k} →
    36 p.toOuterMeasure (R k) ≤ δ * p.toOuterMeasure {o | f o = k}) :
    37 p.toOuterMeasure ({o | p.toOuterMeasure {x | f x = f o} < τ} ∪
    38 {o | τ ≤ p.toOuterMeasure {x | f x = f o} ∧ o ∈ R (f o)}) ≤
    39 p.toOuterMeasure {o | p.toOuterMeasure {x | f x = f o} < τ} + δ
    40
    41end Lax342547.MixtureRecovery
    42
    Show ProofShow ProofShow ProofShow Proof
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