| import Mathlib.Probability.ProductMeasure |
| import Mathlib.MeasureTheory.Measure.Prod |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Finset |
| open scoped ENNReal |
| noncomputable section |
|
|
| /-- Replacing one independent coordinate is a measurable pushforward of a |
| single fresh draw and the original full product. -/ |
| theorem infinitePi_update_law {ι X : Type*} [DecidableEq ι] [MeasurableSpace X] |
| (μ : ι → Measure X) [∀ i, IsProbabilityMeasure (μ i)] |
| (i : ι) (ν : Measure X) [IsProbabilityMeasure ν] : |
| (ν.prod (Measure.infinitePi μ)).map (fun z : X × (ι → X) => Function.update z.2 i z.1) = |
| Measure.infinitePi (Function.update μ i ν) := by |
| let : ∀ k, IsProbabilityMeasure (Function.update μ i ν k) := fun k => by |
| by_cases hk : k = i |
| · subst k; simpa using (inferInstance : IsProbabilityMeasure ν) |
| · simpa [Function.update_of_ne hk] using (inferInstance : IsProbabilityMeasure (μ k)) |
| have hm : Measurable (fun z : X × (ι → X) => Function.update z.2 i z.1) := by |
| apply Measurable.of_eval |
| intro k |
| by_cases hk : k = i |
| · subst k; simpa using (measurable_fst : Measurable (Prod.fst : X × (ι → X) → X)) |
| · simpa [Function.update_of_ne hk,Function.comp_def] using (measurable_pi_apply k).comp |
| (measurable_snd : Measurable (Prod.snd : X × (ι → X) → (ι → X))) |
| apply Measure.eq_infinitePi |
| intro S A hA |
| rw [Measure.map_apply hm (MeasurableSet.pi S.countable_toSet (fun k _ => hA k))] |
| by_cases hi : i ∈ S |
| · have he : (fun z : X × (ι → X) => Function.update z.2 i z.1) ⁻¹' Set.pi (S : Set ι) A = |
| A i ×ˢ Set.pi (↑(S.erase i) : Set ι) A := by |
| ext z |
| simp only [Set.mem_preimage,Set.mem_pi,Finset.mem_coe,Set.mem_prod,mem_erase] |
| constructor |
| · intro hz |
| refine ⟨by simpa using hz i hi,?_⟩ |
| intro k hk |
| simpa [Function.update_of_ne hk.1] using hz k hk.2 |
| · rintro ⟨hz,hrest⟩ k hk |
| by_cases hki : k = i |
| · subst k; simpa using hz |
| · simpa [Function.update_of_ne hki] using hrest k ⟨hki,hk⟩ |
| rw [he,Measure.prod_prod,Measure.infinitePi_pi _ (fun k _ => hA k)] |
| rw [← Finset.mul_prod_erase S (fun k => Function.update μ i ν k (A k)) hi] |
| simp only [Function.update_self] |
| congr 1 |
| exact Finset.prod_congr rfl fun k hk => by rw [Function.update_of_ne (mem_erase.mp hk).1] |
| · have he : (fun z : X × (ι → X) => Function.update z.2 i z.1) ⁻¹' Set.pi (S : Set ι) A = |
| Set.univ ×ˢ Set.pi (S : Set ι) A := by |
| ext z |
| simp only [Set.mem_preimage,Set.mem_pi,Finset.mem_coe,Set.mem_prod,Set.mem_univ,true_and] |
| apply forall_congr' |
| intro k |
| constructor <;> intro h hk |
| · simpa [Function.update_of_ne (show k ≠ i from fun he => hi (he ▸ hk))] using h hk |
| · simpa [Function.update_of_ne (show k ≠ i from fun he => hi (he ▸ hk))] using h hk |
| rw [he,Measure.prod_prod,measure_univ,one_mul,Measure.infinitePi_pi _ (fun k _ => hA k)] |
| exact Finset.prod_congr rfl fun k hk => by |
| rw [Function.update_of_ne (show k ≠ i from fun he => hi (he ▸ hk))] |
|
|
| /-- Absolute continuity survives finitely many coordinate changes. No assertion |
| about arbitrary infinite families of absolutely continuous marginals is used. -/ |
| theorem infinitePi_ac_of_finite_changes {ι X : Type*} [DecidableEq ι] [MeasurableSpace X] |
| (μ ν : ι → Measure X) [∀ i, IsProbabilityMeasure (μ i)] |
| [∀ i, IsProbabilityMeasure (ν i)] (F : Finset ι) |
| (hac : ∀ i ∈ F, ν i ≪ μ i) (heq : ∀ i ∉ F, ν i = μ i) : |
| Measure.infinitePi ν ≪ Measure.infinitePi μ := by |
| induction F using Finset.induction_on generalizing ν with |
| | empty => |
| have h : ν = μ := funext fun i => heq i (by simp) |
| rw [h] |
| | @insert i F hi ih => |
| let σ := Function.update ν i (μ i) |
| let : ∀ k, IsProbabilityMeasure (σ k) := fun k => by |
| by_cases hk : k = i |
| · subst k; simpa [σ] using (inferInstance : IsProbabilityMeasure (μ i)) |
| · simpa [σ,Function.update_of_ne hk] using (inferInstance : IsProbabilityMeasure (ν k)) |
| have hσ : Measure.infinitePi σ ≪ Measure.infinitePi μ := by |
| apply ih σ |
| · intro k hk |
| have hki : k ≠ i := fun h => hi (h ▸ hk) |
| simpa [σ,Function.update_of_ne hki] using hac k (mem_insert_of_mem hk) |
| · intro k hk |
| by_cases hki : k = i |
| · subst k; simp [σ] |
| · simpa [σ,Function.update_of_ne hki] using heq k (by simp [hki,hk]) |
| have hupdate : Function.update σ i (ν i) = ν := by |
| ext k |
| by_cases hk : k = i <;> simp [σ,hk,Function.update_of_ne] |
| have hsame : Function.update σ i (μ i) = σ := by simp [σ] |
| have hprod := (hac i (mem_insert_self i F)).prod |
| (Measure.AbsolutelyContinuous.rfl : Measure.infinitePi σ ≪ Measure.infinitePi σ) |
| have hmap : ((ν i).prod (Measure.infinitePi σ)).map (fun z : X × (ι → X) => Function.update z.2 i z.1) ≪ |
| ((μ i).prod (Measure.infinitePi σ)).map (fun z : X × (ι → X) => Function.update z.2 i z.1) := by |
| apply hprod.map |
| apply Measurable.of_eval |
| intro k |
| by_cases hk : k = i |
| · subst k; simpa using (measurable_fst : Measurable (Prod.fst : X × (ι → X) → X)) |
| · simpa [Function.update_of_ne hk,Function.comp_def] using (measurable_pi_apply k).comp |
| (measurable_snd : Measurable (Prod.snd : X × (ι → X) → (ι → X))) |
| rw [infinitePi_update_law,infinitePi_update_law,hupdate,hsame] at hmap |
| exact hmap.trans hσ |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
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|
