Theorems · Theorem · probability

ProbabilityTheory.Kernel.condDistrib_trajMeasure

∀ {X : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (X n)]
  {κ : (n : ℕ) → ProbabilityTheory.Kernel ((i : ↥(Finset.Iic n)) → X ↑i) (X (n + 1))}
  [inst_1 : ∀ (n : ℕ), ProbabilityTheory.IsMarkovKernel (κ n)] {μ₀ : MeasureTheory.Measure (X 0)}
  [inst_2 : MeasureTheory.IsProbabilityMeasure μ₀] {a : ℕ} [inst_3 : StandardBorelSpace (X (a + 1))]
  [inst_4 : Nonempty (X (a + 1))],
  ⇑(ProbabilityTheory.condDistrib (fun x => x (a + 1)) (Preorder.frestrictLe a)
        (ProbabilityTheory.Kernel.trajMeasure μ₀
          κ)) =ᵐ[MeasureTheory.Measure.map (Preorder.frestrictLe a) (ProbabilityTheory.Kernel.trajMeasure μ₀ κ)]
    ⇑(κ a)

A regular conditional probability distribution of the point at time a + 1 given the trajectory up to time a corresponds to the kernel κ a.

Defined in

Mathlib.Probability.Kernel.IonescuTulcea.Traj

Cited by

0 results in Mathlib

Foundations

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

Assumes

MeasurableSpaceProbabilityTheory.IsMarkovKernelMeasureTheory.IsProbabilityMeasureStandardBorelSpaceNonempty