`BookProof.ChapterHierarchicalBayesComposition.threeLevel_outerPosterior_eq_bayesUpdate` (outer : A → ℝ) (k₁ : A → B → ℝ) (k₂ : B → C → ℝ) (likelihood : C → ℝ) :...

Proved

BookProof.ChapterHierarchicalBayesComposition.threeLevel_outerPosterior_eq_bayesUpdate

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterHierarchicalBayesComposition.

BookProof.ChapterHierarchicalBayesComposition.threeLevel_outerPosterior_eq_bayesUpdate (outer : A → ℝ) (k₁ : A → B → ℝ) (k₂ : B → C → ℝ) (likelihood : C → ℝ) : BookProof.ChapterHierarchicalBayes.outerPosterior outer (compKernel k₁ k₂) (fun _ c => likelihood c) = BookProof.ChapterSequentialBayes.bayesUpdate outer (BookProof.ChapterHierarchicalBayes.marginalLikelihood (compKernel k₁ k₂) (fun _ c => likelihood c))

Formalization note: Lean 4 identifier BookProof.ChapterHierarchicalBayesComposition.threeLevel_outerPosterior_eq_bayesUpdate.

Preamble

-- Generated from ChapterHierarchicalBayesComposition.lean — theorem BookProof.ChapterHierarchicalBayesComposition.threeLevel_outerPosterior_eq_bayesUpdate
import Mathlib
import Definitions.Def_ChapterHierarchicalBayesComposition
import Definitions.Def_ChapterHierarchicalBayes
import Definitions.Def_ChapterSequentialBayes
open BookProof.ChapterHierarchicalBayes
open BookProof.ChapterSequentialBayes
open BookProof.ChapterHierarchicalBayesComposition


open scoped BigOperators


variable {A B C D : Type*}
  [Fintype A] [Fintype B] [Fintype C] [Fintype D]
  [DecidableEq A] [DecidableEq B] [DecidableEq C] [DecidableEq D]

Formal statement

theorem BookProof.ChapterHierarchicalBayesComposition.threeLevel_outerPosterior_eq_bayesUpdate (outer : A → ℝ)
    (k₁ : A → B → ℝ) (k₂ : B → C → ℝ) (likelihood : C → ℝ) :
    BookProof.ChapterHierarchicalBayes.outerPosterior outer
        (compKernel k₁ k₂) (fun _ c => likelihood c) =
      BookProof.ChapterSequentialBayes.bayesUpdate outer
        (BookProof.ChapterHierarchicalBayes.marginalLikelihood
          (compKernel k₁ k₂) (fun _ c => likelihood c)) := by sorry

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