Theorems · Theorem · probability

ProbabilityTheory.rnDeriv_posterior

∀ {Ω : Type u_1} {𝓧 : Type u_2} {mΩ : MeasurableSpace Ω} {m𝓧 : MeasurableSpace 𝓧} {κ : ProbabilityTheory.Kernel Ω 𝓧}
  {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsFiniteMeasure μ] [inst_1 : ProbabilityTheory.IsFiniteKernel κ]
  [inst_2 : StandardBorelSpace Ω] [inst_3 : Nonempty Ω] [inst_4 : MeasurableSpace.CountableOrCountablyGenerated Ω 𝓧],
  (∀ᵐ (ω : Ω) ∂μ, (κ ω).AbsolutelyContinuous (μ.bind ⇑κ)) →
    ∀ᵐ (ω : Ω) ∂μ,
      ∀ᵐ (x : 𝓧) ∂μ.bind ⇑κ,
        (ProbabilityTheory.posterior κ μ).rnDeriv (ProbabilityTheory.Kernel.const 𝓧 μ) x ω =
          κ.rnDeriv (ProbabilityTheory.Kernel.const Ω (μ.bind ⇑κ)) ω x

Defined in

Mathlib.Probability.Kernel.Posterior

Cited by

1 results in Mathlib

Foundations

Depth 338 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsFiniteMeasureProbabilityTheory.IsFiniteKernelStandardBorelSpaceNonemptyMeasurableSpace.CountableOrCountablyGenerated