Theorems · Theorem · statistics

ProbabilityTheory.bayesRisk_lt_top

∀ {Θ : Type u_1} {𝓧 : Type u_2} {𝓨 : Type u_4} {mΘ : MeasurableSpace Θ} {m𝓧 : MeasurableSpace 𝓧}
  {m𝓨 : MeasurableSpace 𝓨} {ℓ : Θ → 𝓨 → ENNReal} [Nonempty 𝓨] (P : ProbabilityTheory.Kernel Θ 𝓧)
  [ProbabilityTheory.IsFiniteKernel P] (π : MeasureTheory.Measure Θ) [MeasureTheory.IsFiniteMeasure π] {C : NNReal},
  (∀ (θ : Θ) (y : 𝓨), ℓ θ y ≤ ↑C) → ProbabilityTheory.bayesRisk ℓ P π < ⊤

For a bounded loss, the Bayes risk with respect to a prior is finite.

Defined in

Mathlib.Probability.Decision.Risk.Basic

Cited by

0 results in Mathlib

Foundations

Depth 218 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NonemptyProbabilityTheory.IsFiniteKernelMeasureTheory.IsFiniteMeasure