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Adaptive sampling on disjoint coordinate sets

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

proven

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    Natural Language Statement

    Lemma

    Arbitrary disjoint exposed and tested index sets inherit the conditional product-law power bound through an explicit sum embedding.

    Concept map
    16 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 disjoint_adaptive_bound proven

    2 embedding_adaptive_bound proven

    Lean source view on GitHub

    1import Lax342547.AdaptiveSampling
    2
    3/-!
    4---
    5title: Adaptive sampling on disjoint coordinate sets
    6type: lemma
    7---
    8Arbitrary disjoint exposed and tested index sets inherit the conditional product-law power bound through an explicit sum embedding.
    9-/
    10
    11namespace Lax342547.AdaptiveSets
    12
    13open Lax342547.RelativeEntropy Lax342547.FiniteSampling Lax342547.RetainedImages
    14open scoped BigOperators
    15
    16axiom embedding_adaptive_bound {E J ι Ω : Type} [Fintype E] [Fintype J] [Fintype ι] [Fintype Ω]
    17 [DecidableEq E] [DecidableEq J] [DecidableEq ι]
    18 (μ : Ω → ℝ) (a : E ⊕ J ↪ ι) (A : (E → Ω) → Ω → Prop) (p : ℝ)
    19 (hμ : Probability μ) (hA : ∀ exposed, cellMass μ (A exposed) ≤ p) :
    20 cellMass (productLaw (fun _ : ι => μ))
    21 (fun sample => ∀ j, A (fun e => sample (a (.inl e))) (sample (a (.inr j)))) ≤ p^Fintype.card J
    22
    23noncomputable def disjointEmbedding {ι : Type} [DecidableEq ι] (E J : Finset ι)
    24 (h : Disjoint E J) : E ⊕ J ↪ ι :=
    25 { toFun := Sum.elim Subtype.val Subtype.val
    26 inj' := by
    27 intro x y hxy
    28 cases x with
    29 | inl e =>
    30 cases y with
    31 | inl e' => exact congrArg Sum.inl (Subtype.ext hxy)
    32 | inr j =>
    33 dsimp at hxy
    34 exact False.elim ((Finset.disjoint_left.mp h) e.property (hxy.symm ▸ j.property))
    35 | inr j =>
    36 cases y with
    37 | inl e =>
    38 dsimp at hxy
    39 exact False.elim ((Finset.disjoint_left.mp h) e.property (hxy ▸ j.property))
    40 | inr j' => exact congrArg Sum.inr (Subtype.ext hxy) }
    41
    42axiom disjoint_adaptive_bound {ι Ω : Type} [Fintype ι] [Fintype Ω] [DecidableEq ι]
    43 (μ : Ω → ℝ) (E J : Finset ι) (h : Disjoint E J)
    44 (A : (E → Ω) → Ω → Prop) (p : ℝ)
    45 (hμ : Probability μ) (hA : ∀ exposed, cellMass μ (A exposed) ≤ p) :
    46 cellMass (productLaw (fun _ : ι => μ))
    47 (fun sample => ∀ j ∈ J, A (fun e => sample e.val) (sample j)) ≤ p^J.card
    48
    49end Lax342547.AdaptiveSets
    50
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