Lax Lean Archive

While this submission is a draft, it cannot be used by other submissions.

Exact conditioning costs and recovery of finite probability masses

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

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Theorem

    Conditioning divides event probabilities and point densities by the retained mass. Multiplying a conditional law by that actual mass recovers the original unnormalized restriction pointwise.

    Concept map
    1 concept
    100%
    Proven claimThis conceptDescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 5 statements. Each proof establishes one of them relative to its assumptions.

    1 disjoint_mixture proven

    2 filter_density proven

    3 filter_event proven

    4 filter_event_bound proven

    5 weighted_filter proven

    Lean source view on GitHub

    1import Mathlib.Probability.ProbabilityMassFunction.Constructions
    2import Mathlib.Data.Finset.Union
    3
    4/-!
    5---
    6title: Exact conditioning costs and recovery of finite probability masses
    7type: theorem
    8---
    9Conditioning divides event probabilities and point densities by the
    10retained mass. Multiplying a conditional law by that actual mass recovers
    11the original unnormalized restriction pointwise.
    12-/
    13
    14namespace Lax342547.Conditioning
    15
    16open scoped ENNReal
    17
    18axiom filter_event {Ω : Type} [Fintype Ω] (p : PMF Ω) (S T : Set Ω)
    19 (hS : ∃ o ∈ S, o ∈ p.support) :
    20 (p.filter S hS).toOuterMeasure T = p.toOuterMeasure (S ∩ T) / p.toOuterMeasure S
    21
    22axiom filter_event_bound {Ω : Type} [Fintype Ω] (p : PMF Ω) (S T : Set Ω)
    23 (hS : ∃ o ∈ S, o ∈ p.support) (c : ℝ≥0∞)
    24 (hc : p.toOuterMeasure (S ∩ T) ≤ c * p.toOuterMeasure S) :
    25 (p.filter S hS).toOuterMeasure T ≤ c
    26
    27axiom filter_density {Ω : Type} (p μ : PMF Ω) (S : Set Ω)
    28 (hS : ∃ o ∈ S, o ∈ p.support) (c : ℝ≥0∞) (hc : ∀ o, p o ≤ c * μ o) :
    29 ∀ o, (p.filter S hS) o ≤ (c / p.toOuterMeasure S) * μ o
    30
    31axiom weighted_filter {Ω : Type} (p : PMF Ω) (S : Set Ω)
    32 (hS : ∃ o ∈ S, o ∈ p.support) (o : Ω) :
    33 p.toOuterMeasure S * (p.filter S hS) o = S.indicator p o
    34
    35axiom disjoint_mixture {Ω J : Type} [DecidableEq Ω] (p : PMF Ω)
    36 (L : Finset J) (S : J → Finset Ω)
    37 (hdisjoint : ∀ i ∈ L, ∀ j ∈ L, i ≠ j → Disjoint (S i) (S j))
    38 (hS : ∀ j ∈ L, ∃ o ∈ (S j : Set Ω), o ∈ p.support) (o : Ω) :
    39 (∑ j : {j // j ∈ L}, p.toOuterMeasure (S j.val) * (p.filter (S j.val) (hS j.val j.property)) o) =
    40 (L.biUnion S : Set Ω).indicator p o
    41
    42end Lax342547.Conditioning
    43
    Show ProofShow ProofShow ProofShow ProofShow Proof
    Builds on

    none

    Used by
    From Mathlib

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…