File size: 3,771 Bytes
f61e547 | 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 | import OAI.Probability.FreeIsing.Conditioning.DiscreteConditioning
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
theorem indepFun_prodMk_left_of_joint
{Ω A B C : Type*} [MeasurableSpace Ω] [MeasurableSpace A]
[MeasurableSpace B] [MeasurableSpace C]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{X : Ω → A} {Y : Ω → B} {Z : Ω → C}
(hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)
(hXYZ : IndepFun X (fun ω => (Y ω, Z ω)) μ)
(hYZ : IndepFun Y Z μ) :
IndepFun (fun ω => (X ω, Y ω)) Z μ := by
have hXY : IndepFun X Y μ := hXYZ.comp measurable_id measurable_fst
have hmapX : IsProbabilityMeasure (μ.map X) := inferInstance
have hmapY : IsProbabilityMeasure (μ.map Y) := inferInstance
have hmapZ : IsProbabilityMeasure (μ.map Z) := inferInstance
apply (indepFun_iff_map_prod_eq_prod_map_map (hX.prodMk hY).aemeasurable
hZ.aemeasurable).mpr
calc
_ = (μ.map (fun ω => (X ω, Y ω, Z ω))).map
MeasurableEquiv.prodAssoc.symm := by
rw [Measure.map_map (by fun_prop) (hX.prodMk (hY.prodMk hZ))]
rfl
_ = ((μ.map X).prod ((μ.map Y).prod (μ.map Z))).map
MeasurableEquiv.prodAssoc.symm := by
rw [hXYZ.map_prod_eq_prod_map_map hX.aemeasurable (hY.prodMk hZ).aemeasurable,
hYZ.map_prod_eq_prod_map_map hY.aemeasurable hZ.aemeasurable]
_ = ((μ.map X).prod (μ.map Y)).prod (μ.map Z) :=
(measurePreserving_prodAssoc (μ.map X) (μ.map Y) (μ.map Z)).symm.map_eq
_ = _ := by
rw [hXY.map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable]
theorem condIndepFun_prodMk_left_of_joint
{Ω S A B C : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω]
[MeasurableSpace S] [MeasurableSingletonClass S] [Countable S]
[MeasurableSpace A] [MeasurableSpace B] [MeasurableSpace C]
{μ : Measure Ω} [IsFiniteMeasure μ]
{R : Ω → S} {X : Ω → A} {Y : Ω → B} {Z : Ω → C}
(hR : Measurable R) (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)
(hXYZ : CondIndepFun (MeasurableSpace.comap R inferInstance) hR.comap_le
X (fun ω => (Y ω, Z ω)) μ)
(hYZ : CondIndepFun (MeasurableSpace.comap R inferInstance) hR.comap_le Y Z μ) :
CondIndepFun (MeasurableSpace.comap R inferInstance) hR.comap_le
(fun ω => (X ω, Y ω)) Z μ := by
apply (condIndepFun_iff_indepFun_cond_fibers hR (hX.prodMk hY) hZ).mpr
intro s hs
let : IsProbabilityMeasure μ[|R ⁻¹' {s}] := cond_isProbabilityMeasure hs
exact indepFun_prodMk_left_of_joint hX hY hZ
(indepFun_cond_fiber_of_condIndepFun hR hX (hY.prodMk hZ) hXYZ s)
(indepFun_cond_fiber_of_condIndepFun hR hY hZ hYZ s)
theorem condIndepFun_coordinate_others
{Ω ι : Type*} [mΩ : MeasurableSpace Ω] [StandardBorelSpace Ω]
{B : ι → Type*} [mB : ∀ i, MeasurableSpace (B i)]
{μ : Measure Ω} [IsFiniteMeasure μ]
{m : MeasurableSpace Ω} {hm : m ≤ mΩ} {F : ∀ i, Ω → B i}
(hF : ∀ i, @Measurable Ω (B i) mΩ (mB i) (F i)) (hI : iCondIndepFun m hm F μ) (c : ι) :
CondIndepFun m hm (F c) (fun ω => fun i : {i : ι // i ≠ c} => F i.val ω) μ := by
let : MeasurableSpace Ω := mΩ
have hle : ∀ i : ι, MeasurableSpace.comap (F i) (mB i) ≤ mΩ :=
fun i => (hF i).comap_le
have hd : Disjoint ({c} : Set ι) {i | i ≠ c} := by simp
have h := condIndep_iSup_of_disjoint hle hI hd
change CondIndep m
(MeasurableSpace.comap (F c) (mB c))
(MeasurableSpace.comap (fun ω => fun i : {i : ι // i ≠ c} => F i.val ω)
inferInstance) hm μ
rw [MeasurableSpace.comap_process_pi, iSup_subtype]
simpa only [Set.mem_singleton_iff, Set.mem_ofPred_eq, iSup_iSup_eq_left] using h
end Problem367
end
end OAI
|