File size: 4,194 Bytes
67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | import Mathlib.Probability.ProductMeasure
import Mathlib.Probability.Independence.InfinitePi
import Mathlib.Probability.Independence.Integration
import Mathlib.Probability.Kernel.Composition.MeasureCompProd
namespace OAI
namespace FiniteFactor
open MeasureTheory ProbabilityTheory Finset MeasurableSpace
open scoped ENNReal
noncomputable section
/-- Products of Markov kernels, with measurability proved on the generating
finite cylinders and extended to the entire product sigma algebra. -/
def independentKernel {ι : Type*} {X Y : ι → Type*}
[∀ i, MeasurableSpace (X i)] [∀ i, MeasurableSpace (Y i)]
(κ : ∀ i, Kernel (X i) (Y i)) [∀ i, IsMarkovKernel (κ i)] :
Kernel (∀ i, X i) (∀ i, Y i) where
toFun x := Measure.infinitePi (fun i => κ i (x i))
measurable' := by
apply Measurable.measure_of_isPiSystem_of_isProbabilityMeasure
generateFrom_squareCylinders.symm
(isPiSystem_squareCylinders (fun _ => isPiSystem_measurableSet) (by simp))
rintro _ ⟨I, S, hS, rfl⟩
have hs (i : ι) : MeasurableSet (S i) := hS i (Set.mem_univ i)
simp_rw [Measure.infinitePi_pi _ (fun i _ => hs i)]
exact Finset.measurable_prod _ fun i _ => ((κ i).measurable_coe (hs i)).comp (measurable_pi_apply i)
instance independentKernel_markov {ι : Type*} {X Y : ι → Type*}
[∀ i, MeasurableSpace (X i)] [∀ i, MeasurableSpace (Y i)]
(κ : ∀ i, Kernel (X i) (Y i)) [∀ i, IsMarkovKernel (κ i)] :
IsMarkovKernel (independentKernel κ) := ⟨fun x => by
change IsProbabilityMeasure (Measure.infinitePi (fun i => κ i (x i)))
infer_instance⟩
theorem independentKernel_pi {ι : Type*} {X Y : ι → Type*}
[∀ i, MeasurableSpace (X i)] [∀ i, MeasurableSpace (Y i)]
(κ : ∀ i, Kernel (X i) (Y i)) [∀ i, IsMarkovKernel (κ i)]
(x : ∀ i, X i) (I : Finset ι) (S : ∀ i, Set (Y i))
(hS : ∀ i ∈ I, MeasurableSet (S i)) :
independentKernel κ x (Set.pi (I : Set ι) S) = ∏ i ∈ I, κ i (x i) (S i) :=
Measure.infinitePi_pi _ hS
theorem lintegral_infinitePi_finset_prod {ι : Type*} {X : ι → Type*}
[∀ i, MeasurableSpace (X i)] (μ : ∀ i, Measure (X i)) [∀ i, IsProbabilityMeasure (μ i)]
(I : Finset ι) (f : ∀ i, X i → ℝ≥0∞) (hf : ∀ i, Measurable (f i)) :
(∫⁻ x, ∏ i ∈ I, f i (x i) ∂Measure.infinitePi μ) = ∏ i ∈ I, ∫⁻ t, f i t ∂μ i := by
rw [lintegral_prod_eq_prod_lintegral_of_indepFun I
(fun (i : ι) (x : ∀ i, X i) => f i (x i)) (iIndepFun_infinitePi (P := μ) (X := f) hf)
(fun i => (hf i).comp (measurable_pi_apply i))]
apply Finset.prod_congr rfl
intro i _
exact (measurePreserving_eval_infinitePi μ i).lintegral_comp (hf i)
/-- Full-vector disintegration commutes with an arbitrary independent product.
The equality is of measures on all measurable sets, not just rectangles. -/
theorem independentKernel_compProd {ι : Type*} {X Y : ι → Type*}
[∀ i, MeasurableSpace (X i)] [∀ i, MeasurableSpace (Y i)]
(μ : ∀ i, Measure (X i)) [∀ i, IsProbabilityMeasure (μ i)]
(κ : ∀ i, Kernel (X i) (Y i)) [∀ i, IsMarkovKernel (κ i)] :
((Measure.infinitePi μ) ⊗ₘ (independentKernel κ)).map
(fun z i => (z.1 i, z.2 i)) = Measure.infinitePi (fun i => (μ i) ⊗ₘ (κ i)) := by
have hm : Measurable (fun z : (∀ i, X i) × (∀ i, Y i) => fun i => (z.1 i, z.2 i)) := by
fun_prop
apply Measure.eq_infinitePi
intro I S hS
rw [Measure.map_apply hm (MeasurableSet.pi I.countable_toSet (fun i _ => hS i)),
Measure.compProd_apply (hm (MeasurableSet.pi I.countable_toSet (fun i _ => hS i)))]
have he (x : ∀ i, X i) : Prod.mk x ⁻¹' ((fun (z : (∀ i, X i) × (∀ i, Y i)) i => (z.1 i, z.2 i)) ⁻¹' Set.pi (I : Set ι) S) =
Set.pi (I : Set ι) (fun i => Prod.mk (x i) ⁻¹' S i) := rfl
simp_rw [he, independentKernel_pi κ _ I _ (fun i _ => measurable_prodMk_left (hS i))]
rw [lintegral_infinitePi_finset_prod μ I
(fun i t => κ i t (Prod.mk t ⁻¹' S i)) (fun i => Kernel.measurable_kernel_prodMk_left (hS i))]
apply Finset.prod_congr rfl
intro i _
exact (Measure.compProd_apply (hS i)).symm
end
end FiniteFactor
end OAI
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