Prove2Me

`BookProof.ChapterHierarchicalBayes.outerPosterior_eq_bayesUpdate` (outer : A → ℝ) (inner : A → B → ℝ) (likelihood : A → B → ℝ) : outerPosterior outer inner likelihood = BookProof.

Proved
BookProof.ChapterHierarchicalBayes.outerPosterior_eq_bayesUpdate

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterHierarchicalBayes.

BookProof.ChapterHierarchicalBayes.outerPosterior_eq_bayesUpdate (outer : A → ℝ) (inner : A → B → ℝ) (likelihood : A → B → ℝ) : outerPosterior outer inner likelihood = BookProof.ChapterSequentialBayes.bayesUpdate outer (marginalLikelihood inner likelihood)

Formalization note: Lean 4 identifier BookProof.ChapterHierarchicalBayes.outerPosterior_eq_bayesUpdate.

Preamble
-- Generated from ChapterHierarchicalBayes.lean — theorem BookProof.ChapterHierarchicalBayes.outerPosterior_eq_bayesUpdate
import Mathlib
import Definitions.Def_ChapterHierarchicalBayes
import Definitions.Def_ChapterSequentialBayes
open BookProof.ChapterSequentialBayes
open BookProof.ChapterHierarchicalBayes


open scoped BigOperators


variable {A B : Type*} [Fintype A] [Fintype B]
Formal statement
theorem BookProof.ChapterHierarchicalBayes.outerPosterior_eq_bayesUpdate (outer : A → ℝ) (inner : A → B → ℝ)
    (likelihood : A → B → ℝ) :
    outerPosterior outer inner likelihood =
      BookProof.ChapterSequentialBayes.bayesUpdate outer
        (marginalLikelihood inner likelihood) := by sorry

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