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67aee1d | 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 | import Mathlib
import OAI.Analysis.HardSphere.Comparison.ConditionalLikelihood
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped ENNReal
open MeasureTheory Function Set Filter
open scoped ENNReal BigOperators
namespace HardSphere.Comparison
open ProbabilityTheory
section ReferenceMeasure
variable {ι : Type*} [DecidableEq ι] {X : ι → Type*}
[∀ i, MeasurableSpace (X i)] {R : Type*} [Fintype R]
(reference : R → LocalCheck X)
(μ : ∀ i, Measure (X i)) [∀ i, IsProbabilityMeasure (μ i)]
omit [∀ i, IsProbabilityMeasure (μ i)] in
lemma lintegral_referenceMeasure (s : Finset ι) (x : ∀ i, X i)
(f : (∀ i, X i) → ℝ≥0∞) (hf : Measurable f) :
(∫⁻ u, f (Function.updateFinset x s u) ∂referenceMeasure reference μ s x) =
lmarginal μ s (fun y ↦ checkWeight reference s y * f y) x /
checkNormalizer reference μ s x := by
change (∫⁻ u, f (Function.updateFinset x s u)
∂(Measure.pi (fun i : s ↦ μ i)).withDensity
(fun u ↦ checkWeight reference s (Function.updateFinset x s u) /
checkNormalizer reference μ s x)) = _
have hD : Measurable (fun u : ∀ i : s, X i ↦
checkWeight reference s (Function.updateFinset x s u) / checkNormalizer reference μ s x) :=
((measurable_checkWeight reference s).comp measurable_updateFinset).div_const _
have hF : Measurable (fun u : ∀ i : s, X i ↦ f (Function.updateFinset x s u)) :=
hf.comp measurable_updateFinset
rw [lintegral_withDensity_eq_lintegral_mul (Measure.pi (fun i : s ↦ μ i)) hD hF]
simp only [Pi.mul_apply, div_eq_mul_inv]
simp_rw [mul_right_comm _ _ (f _)]
exact lintegral_mul_const _
(((measurable_checkWeight reference s).mul hf).comp measurable_updateFinset)
end ReferenceMeasure
variable {ι : Type*} [DecidableEq ι] {X Z : ι → Type*}
[∀ i, MeasurableSpace (X i)] [∀ i, MeasurableSpace (Z i)]
{R C : Type*} [Fintype R] [Fintype C]
(reference : R → LocalCheck X) (checks : C → LocalCheck (fun i ↦ X i × Z i))
(μ : ∀ i, Measure (X i)) [∀ i, IsProbabilityMeasure (μ i)]
(κ : ∀ i, Kernel (X i) (Z i)) [∀ i, IsFiniteKernel (κ i)]
theorem conditionalLikelihood_mean (s o : Finset ι) (hso : Disjoint s o)
{a : ℝ≥0∞} (ha : 0 < a)
(hκm : ∀ i, ∫⁻ x, κ i x Set.univ ∂μ i = 1)
(hRef : UniformInsertionOn reference μ a (s ∪ o))
(hcompat : UniformInsertionOn checks (fun i ↦ (μ i).compProd (κ i)) a s)
(hOutside : ∀ (x : ∀ i, X i) (z : ∀ i, Z i), x ∈ retainedEvent reference o →
(fun i ↦ (x i,z i)) ∈ retainedEvent checks o)
(hincl : ∀ (x : ∀ i, X i) (z : ∀ i, Z i),
checkWeight checks (s ∪ o) (fun i ↦ (x i,z i)) ≤ checkWeight reference (s ∪ o) x)
(x : ∀ i, X i) (z : ∀ i, Z i) :
(∫⁻ u, conditionalLikelihood reference checks μ κ s o
(Function.updateFinset x (s ∪ o) u) z ∂referenceMeasure reference μ (s ∪ o) x) = 1 := by
rw [lintegral_referenceMeasure reference μ (s ∪ o) x _
(measurable_conditionalLikelihood reference checks μ κ s o z)]
have hm := (measurable_checkWeight reference (s ∪ o)).mul
(measurable_conditionalLikelihood reference checks μ κ s o z)
have heq : lmarginal μ s (fun y ↦ checkWeight reference (s ∪ o) y *
conditionalLikelihood reference checks μ κ s o y z) =
partialNormalizer reference μ s o := by
funext y
exact conditionalLikelihood_marginal reference checks μ κ s o hso ha hκm hcompat
hOutside hincl y z
rw [lmarginal_union' μ (fun y ↦ checkWeight reference (s ∪ o) y *
conditionalLikelihood reference checks μ κ s o y z) hm hso, heq]
have hZ : lmarginal μ o (partialNormalizer reference μ s o) =
checkNormalizer reference μ (s ∪ o) := by
exact (lmarginal_union' μ (checkWeight reference (s ∪ o))
(measurable_checkWeight reference (s ∪ o)) hso).symm
rw [hZ]
exact ENNReal.div_self (ne_of_gt (checkNormalizer_pos_on reference ha (s ∪ o) hRef x))
(ne_top_of_le_ne_top ENNReal.one_ne_top (checkNormalizer_le_one reference (s ∪ o) x))
end HardSphere.Comparison
end
end OAI
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