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import OAI.Probability.Ballisticity.Model
namespace OAI
section
open MeasureTheory ProbabilityTheory InformationTheory
open scoped ENNReal Classical
namespace DirectionalTransience.StoppedWindow
variable {Z F D V : Type*} [MeasurableSpace Z] [MeasurableSpace F]
[MeasurableSpace D] [MeasurableSpace V]
lemma product_map_reference (ζ : Measure Z) [IsFiniteMeasure ζ]
(π : Measure F) [IsProbabilityMeasure π]
(data : Z → D) (hd : Measurable data)
(field : Z → F → V) (hf : Measurable (fun p : Z × F => field p.1 p.2))
(R : Kernel D V) [IsMarkovKernel R]
(hR : ∀ z, π.map (field z) = R (data z)) :
(ζ.prod π).map (fun p => (data p.1,field p.1 p.2)) = (ζ.map data).compProd R := by
have hout : Measurable (fun p : Z × F => (data p.1,field p.1 p.2)) :=
(hd.comp measurable_fst).prodMk hf
apply Measure.ext_prod
intro A B hA hB
rw [Measure.map_apply hout (hA.prod hB),
Measure.prod_apply ((hA.prod hB).preimage hout),
Measure.compProd_apply_prod hA hB, ← lintegral_indicator hA,
lintegral_map (by exact (R.measurable_coe hB).indicator hA) hd]
apply lintegral_congr
intro z
by_cases hz : data z ∈ A
· rw [Set.indicator_of_mem hz]
have he : Prod.mk z ⁻¹' ((fun p : Z × F => (data p.1,field p.1 p.2)) ⁻¹' (A ×ˢ B)) =
field z ⁻¹' B := by ext v; simp [hz]
have hfz : Measurable (field z) := hf.comp measurable_prodMk_left
rw [he, ← Measure.map_apply hfz hB,hR z]
· have he : Prod.mk z ⁻¹' ((fun p : Z × F => (data p.1,field p.1 p.2)) ⁻¹' (A ×ˢ B)) = ∅ := by
ext v; simp [hz]
simp [he,hz]
lemma map_data_field_fst (μ : Measure (Z × F))
(data : Z → D) (hd : Measurable data)
(field : Z → F → V) (hf : Measurable (fun p : Z × F => field p.1 p.2)) :
(μ.map (fun p => (data p.1,field p.1 p.2))).fst = μ.fst.map data := by
calc
_ = μ.map (fun p : Z × F => data p.1) :=
Measure.fst_map_prodMk (hd.comp measurable_fst) hf
_ = _ := (Measure.map_map hd measurable_fst).symm
theorem own_marginal_projection (μ : Measure (Z × F)) [IsFiniteMeasure μ]
(π : Measure F) [IsProbabilityMeasure π]
(data : Z → D) (hd : Measurable data)
(field : Z → F → V) (hf : Measurable (fun p : Z × F => field p.1 p.2))
(R : Kernel D V) [IsMarkovKernel R]
(hR : ∀ z, π.map (field z) = R (data z)) :
let out := μ.map (fun p => (data p.1,field p.1 p.2))
klDiv out (out.fst.compProd R) ≤ klDiv μ (μ.fst.prod π) := by
have hout : Measurable (fun p : Z × F => (data p.1,field p.1 p.2)) :=
(hd.comp measurable_fst).prodMk hf
have h := klDiv_map_le μ (μ.fst.prod π) hout
rw [product_map_reference μ.fst π data hd field hf R hR] at h
dsimp only
rw [map_data_field_fst μ data hd field hf]
exact h
end DirectionalTransience.StoppedWindow
end
end OAI
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