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ddc84c5 | 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 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | 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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