File size: 3,199 Bytes
9eeb9c7 | 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 | import Mathlib.Probability.Independence.Conditional
import Mathlib.Probability.Independence.Basic
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped BigOperators ProbabilityTheory
namespace Invariant233.ConditionalDistributionEvents
variable {ι Ω X Y : Type*} [Countable ι]
[MeasurableSpace Ω] [StandardBorelSpace Ω]
[MeasurableSpace X] [MeasurableSpace Y] [StandardBorelSpace Y] [Nonempty Y]
theorem iCondIndepSet_preimage_of_condDistrib
(μ : Measure Ω) [IsFiniteMeasure μ]
(observed : Ω → X) (value : Ω → Y)
(hobserved : Measurable observed) (hvalue : Measurable value)
(events : ι → Set Y) (hevents : ∀ i, MeasurableSet (events i))
(hind : ∀ᵐ x ∂μ.map observed,
iIndepSet events (condDistrib value observed μ x)) :
iCondIndepSet (MeasurableSpace.comap observed inferInstance) hobserved.comap_le
(fun i => value ⁻¹' events i) μ := by
apply (iCondIndepSet_iff _ _ _ (fun i => hvalue (hevents i)) μ).mpr
intro S
have hi : MeasurableSet (⋂ i ∈ S, events i) :=
MeasurableSet.biInter S.countable_toSet (fun i _ => hevents i)
have hfull := ae_of_ae_map hobserved.aemeasurable hind
have heach : ∀ᵐ ω ∂μ, ∀ i,
(condDistrib value observed μ (observed ω)).real (events i) =
(μ⟦value ⁻¹' events i | MeasurableSpace.comap observed inferInstance⟧) ω :=
ae_all_iff.mpr (fun i => condDistrib_ae_eq_condExp hobserved hvalue (hevents i))
filter_upwards [hfull, heach, condDistrib_ae_eq_condExp hobserved hvalue hi]
with ω hω heach hinter
have hprod := (iIndepSet_iff_meas_biInter hevents).mp hω S
have hreal := congrArg ENNReal.toReal hprod
simp only [ENNReal.toReal_prod] at hreal
have hpre : value ⁻¹' (⋂ i ∈ S, events i) = ⋂ i ∈ S, value ⁻¹' events i := by
ext z
simp
rw [hpre] at hinter
simp only [Finset.prod_apply]
rw [← hinter]
change (condDistrib value observed μ (observed ω) (⋂ i ∈ S, events i)).toReal = _
rw [hreal]
exact Finset.prod_congr rfl (fun i _ => heach i)
theorem iCondIndepSet_coordinate_preimage_of_condDistrib
{Z : ι → Type*} [∀ i, MeasurableSpace (Z i)]
(μ : Measure Ω) [IsFiniteMeasure μ]
(observed : Ω → X) (value : Ω → Y)
(hobserved : Measurable observed) (hvalue : Measurable value)
(coordinate : ∀ i, Y → Z i) (hcoordinate : ∀ i, Measurable (coordinate i))
(events : ∀ i, Set (Z i)) (hevents : ∀ i, MeasurableSet (events i))
(hind : ∀ᵐ x ∂μ.map observed,
iIndepFun coordinate (condDistrib value observed μ x)) :
iCondIndepSet (MeasurableSpace.comap observed inferInstance) hobserved.comap_le
(fun i => (fun ω => coordinate i (value ω)) ⁻¹' events i) μ := by
apply iCondIndepSet_preimage_of_condDistrib μ observed value hobserved hvalue
(fun i => coordinate i ⁻¹' events i) (fun i => hcoordinate i (hevents i))
filter_upwards [hind] with x hx
apply (iIndepSet_iff_meas_biInter (fun i => hcoordinate i (hevents i))).mpr
intro S
exact hx.measure_inter_preimage_eq_mul S (fun i _ => hevents i)
end Invariant233.ConditionalDistributionEvents
end
end OAI
|