Theorems · Theorem · statistics

ProbabilityTheory.bayesRisk_le_bayesRisk_comp

∀ {Θ : Type u_1} {𝓧 : Type u_2} {𝓧' : Type u_3} {𝓨 : Type u_4} {mΘ : MeasurableSpace Θ} {m𝓧 : MeasurableSpace 𝓧}
  {m𝓧' : MeasurableSpace 𝓧'} {m𝓨 : MeasurableSpace 𝓨} (ℓ : Θ → 𝓨 → ENNReal) (P : ProbabilityTheory.Kernel Θ 𝓧)
  (π : MeasureTheory.Measure Θ) (η : ProbabilityTheory.Kernel 𝓧 𝓧') [ProbabilityTheory.IsMarkovKernel η],
  ProbabilityTheory.bayesRisk ℓ P π ≤ ProbabilityTheory.bayesRisk ℓ (η.comp P) π

Data processing inequality for the Bayes risk with respect to a prior: composition of the data generating kernel by a Markov kernel increases the risk.

Defined in

Mathlib.Probability.Decision.Risk.Basic

Cited by

2 results in Mathlib

Foundations

Depth 217 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ProbabilityTheory.IsMarkovKernel