| import Mathlib |
| import OAI.Probability.Ballisticity.Model |
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| namespace OAI |
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| section |
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|
| 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] |
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| 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 |
|
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| 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 |
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| end DirectionalTransience.StoppedWindow |
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| end |
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| end OAI |
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