Theorems · Theorem · statistics

ProbabilityTheory.lintegral_iInf_posterior_le_avgRisk

∀ {Θ : Type u_1} {𝓧 : Type u_2} {𝓨 : Type u_3} {mΘ : MeasurableSpace Θ} {m𝓧 : MeasurableSpace 𝓧}
  {m𝓨 : MeasurableSpace 𝓨} {ℓ : Θ → 𝓨 → ENNReal} [inst : StandardBorelSpace Θ] [inst_1 : Nonempty Θ],
  Measurable (Function.uncurry ℓ) →
    ∀ (P : ProbabilityTheory.Kernel Θ 𝓧) [inst_2 : ProbabilityTheory.IsFiniteKernel P]
      (κ : ProbabilityTheory.Kernel 𝓧 𝓨) [ProbabilityTheory.IsMarkovKernel κ] (π : MeasureTheory.Measure Θ)
      [inst_4 : MeasureTheory.IsFiniteMeasure π],
      ∫⁻ (x : 𝓧), ⨅ y, ∫⁻ (θ : Θ), ℓ θ y ∂(ProbabilityTheory.posterior P π) x ∂π.bind ⇑P ≤
        ProbabilityTheory.avgRisk ℓ P κ π

Defined in

Mathlib.Probability.Decision.BayesEstimator

Cited by

1 results in Mathlib

Foundations

Depth 283 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

StandardBorelSpaceNonemptyProbabilityTheory.IsFiniteKernelProbabilityTheory.IsMarkovKernelMeasureTheory.IsFiniteMeasure