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Sampling after exposed coordinates

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

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    Natural Language Statement

    Lemma

    For independent product samples, a cylinder event defined by exposed coordinates has the exact averaged power probability and the uniform power bound.

    Concept map
    15 concepts
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    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 adaptive_cylinder_bound proven

    2 adaptive_cylinder_probability proven

    Lean source view on GitHub

    1import Lax342547.CoordinateRestrictions
    2
    3/-!
    4---
    5title: Sampling after exposed coordinates
    6type: lemma
    7---
    8For independent product samples, a cylinder event defined by exposed coordinates has the exact averaged power probability and the uniform power bound.
    9-/
    10
    11namespace Lax342547.AdaptiveSampling
    12
    13open Lax342547.RelativeEntropy Lax342547.FiniteSampling Lax342547.RetainedImages
    14open scoped BigOperators
    15
    16axiom adaptive_cylinder_probability {J K Ω : Type} [Fintype J] [Fintype K] [Fintype Ω]
    17 [DecidableEq J] [DecidableEq K] (μ : Ω → ℝ) (A : (J → Ω) → Ω → Prop) :
    18 cellMass (productLaw (fun _ : J ⊕ K => μ))
    19 (fun sample => ∀ k, A (fun j => sample (.inl j)) (sample (.inr k))) =
    20 ∑ exposed : J → Ω, productLaw (fun _ : J => μ) exposed*(cellMass μ (A exposed))^Fintype.card K
    21
    22axiom adaptive_cylinder_bound {J K Ω : Type} [Fintype J] [Fintype K] [Fintype Ω]
    23 [DecidableEq J] [DecidableEq K] (μ : Ω → ℝ) (A : (J → Ω) → Ω → Prop) (p : ℝ)
    24 (hμ : Probability μ) (hp : ∀ exposed, cellMass μ (A exposed) ≤ p) :
    25 cellMass (productLaw (fun _ : J ⊕ K => μ))
    26 (fun sample => ∀ k, A (fun j => sample (.inl j)) (sample (.inr k))) ≤ p^Fintype.card K
    27
    28end Lax342547.AdaptiveSampling
    29
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    From Mathlib

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