`BookProof.ChapterBayesInference.posterior_eq_born_conditional` (hprior : ∀ x, 0 ≤ prior x) (hL : ∀ x y, 0 ≤ L x y) (y : Y) (x : X) : posterior prior L y x = (Real.sqrt (joint prio

Proved

BookProof.ChapterBayesInference.posterior_eq_born_conditional

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterBayesInference.

BookProof.ChapterBayesInference.posterior_eq_born_conditional (hprior : ∀ x, 0 ≤ prior x) (hL : ∀ x y, 0 ≤ L x y) (y : Y) (x : X) : posterior prior L y x = (Real.sqrt (joint prior L x y)) ^ 2 / ∑ x', (Real.sqrt (joint prior L x' y)) ^ 2

Formalization note: Lean 4 identifier BookProof.ChapterBayesInference.posterior_eq_born_conditional.

Preamble

-- Generated from ChapterBayesInference.lean — theorem BookProof.ChapterBayesInference.posterior_eq_born_conditional
import Mathlib
import Definitions.Def_ChapterBayesInference
open BookProof.ChapterBayesInference


open scoped BigOperators


variable {X Y : Type*} [Fintype X] [Fintype Y] [DecidableEq X] [DecidableEq Y]

variable {prior : X → ℝ} {L : X → Y → ℝ}

Formal statement

theorem BookProof.ChapterBayesInference.posterior_eq_born_conditional (hprior : ∀ x, 0 ≤ prior x) (hL : ∀ x y, 0 ≤ L x y)
    (y : Y) (x : X) :
    posterior prior L y x =
      (Real.sqrt (joint prior L x y)) ^ 2 / ∑ x', (Real.sqrt (joint prior L x' y)) ^ 2 := by sorry

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