Lax Lean Archive

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

Full feasible support and finite information projection

Lax342547.EntropyProjection · concepts/Lax342547/EntropyProjection.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

    Lemma

    A compact convex feasible family has an entropy minimizer positive on the union of all feasible supports. The exact one-sided variational inequality gives the information-projection bound and the negative log support cost, closing the finite content of Lemma 3.2.

    Concept map
    7 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants 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 entropy_projection proven

    2 minimizer_full_support proven

    3 minimizer_variational proven

    4 projection_inequality proven

    5 projection_support_cost proven

    Lean source view on GitHub

    1import Lax342547.EntropyLines
    2
    3/-!
    4---
    5title: Full feasible support and finite information projection
    6type: lemma
    7---
    8A compact convex feasible family has an entropy minimizer positive on the union of all feasible supports. The exact one-sided variational inequality gives the information-projection bound and the negative log support cost, closing the finite content of Lemma 3.2.
    9-/
    10
    11namespace Lax342547.EntropyProjection
    12
    13open Lax342547.RelativeEntropy Lax342547.EntropyLines
    14open scoped Topology BigOperators
    15open Filter
    16
    17axiom minimizer_full_support {Ω : Type} [Fintype Ω]
    18 (P : Set (Ω → ℝ)) (q ρ : Ω → ℝ)
    19 (hprob : ∀ p ∈ P, Probability p) (hconv : Convex ℝ P) (hρ : ρ ∈ P)
    20 (hmin : ∀ p ∈ P, entropy ρ q ≤ entropy p q) :
    21 ∀ ρ' ∈ P, ∀ x, 0 < ρ' x → 0 < ρ x
    22
    23axiom minimizer_variational {Ω : Type} [Fintype Ω]
    24 (P : Set (Ω → ℝ)) (q ρ : Ω → ℝ)
    25 (hprob : ∀ p ∈ P, Probability p) (hconv : Convex ℝ P) (hρ : ρ ∈ P)
    26 (hmin : ∀ p ∈ P, entropy ρ q ≤ entropy p q) (ρ' : Ω → ℝ) (hρ' : ρ' ∈ P) :
    27 0 ≤ ∑ x, (ρ' x-ρ x)*(Real.log (ρ x)-Real.log (q x))
    28
    29axiom projection_inequality {Ω : Type} [Fintype Ω]
    30 (P : Set (Ω → ℝ)) (q ρ : Ω → ℝ)
    31 (hprob : ∀ p ∈ P, Probability p) (hconv : Convex ℝ P) (hρ : ρ ∈ P)
    32 (hmin : ∀ p ∈ P, entropy ρ q ≤ entropy p q) (ρ' : Ω → ℝ) (hρ' : ρ' ∈ P) :
    33 entropy ρ' ρ ≤ entropy ρ' q-entropy ρ q
    34
    35axiom projection_support_cost {Ω : Type} [Fintype Ω]
    36 (P : Set (Ω → ℝ)) (q ρ : Ω → ℝ)
    37 (hprob : ∀ p ∈ P, Probability p) (hconv : Convex ℝ P) (hρ : ρ ∈ P)
    38 (hmin : ∀ p ∈ P, entropy ρ q ≤ entropy p q) (ρ' : Ω → ℝ) (hρ' : ρ' ∈ P)
    39 (S : Ω → Prop) (hS : ∀ x, ¬ S x → ρ' x = 0)
    40 (hmass : 0 < Lax342547.RetainedImages.cellMass ρ S) :
    41 -Real.log (Lax342547.RetainedImages.cellMass ρ S) ≤ entropy ρ' q-entropy ρ q
    42
    43axiom entropy_projection {Ω : Type} [Fintype Ω]
    44 (P : Set (Ω → ℝ)) (q : Ω → ℝ) (hcompact : IsCompact P) (hne : P.Nonempty)
    45 (hconv : Convex ℝ P) (hprob : ∀ p ∈ P, Probability p) :
    46 ∃ ρ ∈ P, (∀ ρ' ∈ P, entropy ρ q ≤ entropy ρ' q) ∧
    47 (∀ ρ' ∈ P, ∀ x, 0 < ρ' x → 0 < ρ x) ∧
    48 (∀ ρ' ∈ P, entropy ρ' ρ ≤ entropy ρ' q-entropy ρ q)
    49
    50end Lax342547.EntropyProjection
    51
    Show ProofShow ProofShow ProofShow ProofShow Proof
    Builds on
    Used by
    From Mathlib

    none

    Discussion

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

    Loading discussion…