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Finite product sampling and coordinate marginals

Lax263307.FiniteProductSampling · concepts/Lax263307/FiniteProductSampling.lean · lax-263307

proven

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    Natural Language Statement

    Theorem

    Independent finite coordinates have total product mass one. Restricting to any injectively selected coordinates recovers the product law on precisely those coordinates. The remaining coordinates sum out to one.

    Concept map
    3 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

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

    1 embedding_marginal proven

    2 normalized proven

    Lean source view on GitHub

    1import Lax263307.PathSampling
    2import Mathlib.Algebra.BigOperators.Ring.Finset
    3import Mathlib.Logic.Equiv.Prod
    4
    5/-!
    6---
    7title: Finite product sampling and coordinate marginals
    8type: theorem
    9---
    10Independent finite coordinates have total product mass one. Restricting to
    11any injectively selected coordinates recovers the product law on precisely
    12those coordinates. The remaining coordinates sum out to one.
    13
    14# Attribution
    15
    16Finite probability algebra for Bonnet–Chang's coupled sampling in Lemma 11.
    17-/
    18
    19namespace Lax263307.FiniteProductSampling
    20
    21variable {ι κ : Type*} [Fintype ι] [Fintype κ] [DecidableEq ι]
    22
    23noncomputable def weight (f : ι → κ → ℝ) (φ : ι → κ) : ℝ := ∏ a, f a (φ a)
    24
    25axiom normalized (f : ι → κ → ℝ) (mass : ∀ a, (∑ x, f a x) = 1) :
    26 (∑ φ : ι → κ, weight f φ) = 1
    27
    28axiom embedding_marginal {k : ℕ} (f : ι → κ → ℝ)
    29 (mass : ∀ a, (∑ x, f a x) = 1) (j : Fin k ↪ ι) (H : (Fin k → κ) → ℝ) :
    30 (∑ φ : ι → κ, weight f φ * H (fun a => φ (j a))) =
    31 ∑ ψ : Fin k → κ, weight (fun a => f (j a)) ψ * H ψ
    32
    33end Lax263307.FiniteProductSampling
    34
    Show ProofShow Proof

    Attribution

    Finite probability algebra for Bonnet–Chang's coupled sampling in Lemma 11.

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    From Mathlib

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