| import Mathlib |
| import OAI.Analysis.HardSphere.Comparison.ConditionalLikelihood |
| import OAI.Analysis.HardSphere.Conditioning.ProductNormalize |
|
|
| namespace OAI |
|
|
| section |
|
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| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace HardSphere.Comparison |
| variable {I : Type*} [DecidableEq I] {X Z : I → Type*} |
| [∀i,MeasurableSpace (X i)] [∀i,MeasurableSpace (Z i)] |
| {E R : Type*} [Fintype E] [Fintype R] |
| (checks : E → LocalCheck (fun i ↦ X i×Z i)) |
| (reference : R → LocalCheck X) |
| (μ : ∀i,Measure (X i)) [∀i,IsProbabilityMeasure (μ i)] |
| (κ : ∀i,Kernel (X i) (Z i)) [∀i,IsFiniteKernel (κ i)] |
|
|
| lemma conditionalDensity_weighted (s o : Finset I) (x : ∀i,X i) (z : ∀i,Z i) |
| (F : (∀i,X i) → ℝ≥0∞) (hF : Measurable F) : |
| lmarginal μ s (fun y ↦ F y * conditionalDensity checks κ s o y z) x = |
| lmarginal (fun i ↦ (μ i).compProd (κ i)) s |
| (fun v ↦ F (fun i ↦ (v i).1)*checkWeight checks (s∪o) v) (fun i ↦ (x i,z i)) := by |
| let f (p : ∀i : s,X i×Z i) := |
| F (Function.updateFinset x s (fun i ↦ (p i).1))* |
| checkWeight checks (s∪o) (Function.updateFinset (fun i ↦ (x i,z i)) s p) |
| have hf : Measurable f := by |
| apply Measurable.mul |
| · apply hF.comp |
| exact measurable_updateFinset.comp (Measurable.of_eval (fun i ↦ measurable_fst.comp (measurable_pi_apply i))) |
| · exact (measurable_checkWeight checks (s∪o)).comp measurable_updateFinset |
| have he (p : ∀i : s,X i×Z i) : |
| (fun i ↦ (Function.updateFinset (fun i ↦ (x i,z i)) s p i).1)= |
| Function.updateFinset x s (fun i ↦ (p i).1) := by |
| funext i |
| by_cases hi : i∈s <;> simp [Function.updateFinset,hi] |
| change (∫⁻y,F (Function.updateFinset x s y)*conditionalDensity checks κ s o (Function.updateFinset x s y) z |
| ∂Measure.pi (fun i : s ↦ μ i))= _ |
| change _=∫⁻p,F (fun i ↦ (Function.updateFinset (fun i ↦ (x i,z i)) s p i).1)* |
| checkWeight checks (s∪o) (Function.updateFinset (fun i ↦ (x i,z i)) s p) |
| ∂Measure.pi (fun i : s ↦ (μ i).compProd (κ i)) |
| simp_rw [he] |
| rw [show (fun p ↦ F (Function.updateFinset x s (fun i ↦ (p i).1))* |
| checkWeight checks (s∪o) (Function.updateFinset (fun i ↦ (x i,z i)) s p))=f from rfl, |
| lintegral_pi_compProd (fun i : s ↦ κ i) (fun i : s ↦ μ i) hf] |
| apply lintegral_congr |
| intro y |
| change F (Function.updateFinset x s y)* |
| (∫⁻u,checkWeight checks (s∪o) |
| (fun i ↦ (Function.updateFinset x s y i,Function.updateFinset z s u i)) |
| ∂Measure.pi (fun i : s ↦ κ i (Function.updateFinset x s y i)))=_ |
| have hμ : (fun i : s ↦ κ i (Function.updateFinset x s y i))=(fun i : s ↦ κ i (y i)) := by |
| funext i |
| simp [Function.updateFinset,i.property] |
| rw [hμ,←lintegral_const_mul _ (by |
| exact (measurable_checkWeight checks (s∪o)).comp |
| (Measurable.of_eval (fun i ↦ measurable_const.prodMk |
| ((measurable_pi_apply i).comp measurable_updateFinset))))] |
| apply lintegral_congr |
| intro u |
| have hp : (fun i ↦ (Function.updateFinset x s y i,Function.updateFinset z s u i))= |
| Function.updateFinset (fun i ↦ (x i,z i)) s (fun i ↦ (y i,u i)) := by |
| funext i |
| by_cases hi : i∈s <;> simp [Function.updateFinset,hi] |
| simp only [f,hp] |
|
|
| omit [Fintype E] [Fintype R] [∀i,IsProbabilityMeasure (μ i)] [∀i,IsFiniteKernel (κ i)] in |
| lemma conditionalLikelihood_update (s o : Finset I) (x : ∀i,X i) (z : ∀i,Z i) (y : ∀i : s,X i) : |
| conditionalLikelihood reference checks μ κ s o (Function.updateFinset x s y) z= |
| (partialNormalizer reference μ s o x / |
| partialNormalizer checks (fun i ↦ (μ i).compProd (κ i)) s o (fun i ↦ (x i,z i)))* |
| conditionalDensity checks κ s o (Function.updateFinset x s y) z := by |
| unfold conditionalLikelihood |
| rw [partialNormalizer_updateFinset] |
| congr 2 |
| apply lmarginal_congr |
| intro i hi |
| simp [Function.updateFinset,hi] |
|
|
| def proposalFiber (s o : Finset I) (x : ∀i,X i) (z : ∀i,Z i) : Measure (∀i : s,X i×Z i) := |
| conditionedMeasure (Measure.pi (fun i : s ↦ (μ i).compProd (κ i))) |
| {p | Function.updateFinset (fun i ↦ (x i,z i)) s p∈retainedEvent checks (s∪o)} |
|
|
| def referenceFiber (s o : Finset I) (x : ∀i,X i) : Measure (∀i : s,X i) := |
| conditionedMeasure (Measure.pi (fun i : s ↦ μ i)) |
| {p | Function.updateFinset x s p∈retainedEvent reference (s∪o)} |
|
|
| lemma fiber_lintegral {V : I → Type*} [∀i,MeasurableSpace (V i)] |
| {C : Type*} [Fintype C] (c : C → LocalCheck V) (ν : ∀i,Measure (V i)) |
| (s o : Finset I) (v : ∀i,V i) (f : (∀i : s,V i) → ℝ≥0∞) : |
| (∫⁻p,f p ∂conditionedMeasure (Measure.pi (fun i : s ↦ ν i)) |
| {p | Function.updateFinset v s p∈retainedEvent c (s∪o)})= |
| (partialNormalizer c ν s o v)⁻¹ * |
| ∫⁻p,f p*checkWeight c (s∪o) (Function.updateFinset v s p) |
| ∂Measure.pi (fun i : s ↦ ν i) := by |
| let A : Set (∀i : s,V i) := {p | Function.updateFinset v s p∈retainedEvent c (s∪o)} |
| have hA : MeasurableSet A := (measurableSet_retainedEvent c (s∪o)).preimage measurable_updateFinset |
| have hmass : (Measure.pi (fun i : s ↦ ν i)) A=partialNormalizer c ν s o v := by |
| change _=∫⁻p,A.indicator (fun _ ↦ (1:ℝ≥0∞)) p ∂Measure.pi (fun i : s ↦ ν i) |
| rw [lintegral_indicator_const hA,one_mul] |
| change (∫⁻p,f p ∂conditionedMeasure _ A)=_ |
| rw [conditionedMeasure,lintegral_smul_measure,hmass] |
| change _⁻¹ *(∫⁻p in A,f p ∂Measure.pi (fun i : s ↦ ν i))=_ |
| rw [←lintegral_indicator hA] |
| congr 1 |
| apply lintegral_congr |
| intro p |
| by_cases hp : p∈A |
| · simp only [Set.indicator_of_mem hp] |
| change f p=f p*A.indicator (fun _ ↦ (1:ℝ≥0∞)) p |
| rw [Set.indicator_of_mem hp,mul_one] |
| · simp only [Set.indicator_of_notMem hp] |
| change 0=f p*A.indicator (fun _ ↦ (1:ℝ≥0∞)) p |
| rw [Set.indicator_of_notMem hp,mul_zero] |
|
|
| lemma proposalFiber_primary (s o : Finset I) (x : ∀i,X i) (z : ∀i,Z i) |
| (hZ : 0<partialNormalizer reference μ s o x) |
| (hincl : ∀(y : ∀i,X i)(u : ∀i,Z i), |
| checkWeight checks (s∪o) (fun i ↦ (y i,u i))≤checkWeight reference (s∪o) y) : |
| (proposalFiber checks μ κ s o x z).map (fun p i ↦ (p i).1)= |
| (referenceFiber reference μ s o x).withDensity |
| (fun y ↦ conditionalLikelihood reference checks μ κ s o (Function.updateFinset x s y) z) := by |
| let Z₀ := partialNormalizer reference μ s o x |
| let Y₀ := partialNormalizer checks (fun i ↦ (μ i).compProd (κ i)) s o (fun i ↦ (x i,z i)) |
| have hZT : Z₀≠⊤ := ne_top_of_le_ne_top ENNReal.one_ne_top |
| (partialNormalizer_le_one reference s o x) |
| have hZm : Z₀⁻¹*Z₀=1 := ENNReal.inv_mul_cancel (ne_of_gt hZ) hZT |
| apply Measure.ext_of_lintegral |
| intro f hf |
| let F (y : ∀i,X i) := f (fun i : s ↦ y i) |
| have hF : Measurable F := hf.comp (Measurable.of_eval (fun i ↦ measurable_pi_apply (i:I))) |
| have hu (p : ∀i : s,X i) : F (Function.updateFinset x s p)=f p := by |
| dsimp only [F] |
| congr 1 |
| funext i |
| simp [Function.updateFinset,i.property] |
| have hm : Measurable (fun y : ∀i:s,X i ↦ conditionalLikelihood reference checks μ κ s o |
| (Function.updateFinset x s y) z) := |
| (measurable_conditionalLikelihood reference checks μ κ s o z).comp |
| (measurable_updateFinset (x:=x) (s:=s)) |
| have hproj : Measurable (fun p : ∀i:s,X i×Z i ↦ fun i ↦ (p i).1) := |
| Measurable.of_eval (fun i ↦ measurable_fst.comp (measurable_pi_apply i)) |
| rw [lintegral_map hf hproj, |
| lintegral_withDensity_eq_lintegral_mul _ hm hf] |
| simp only [Pi.mul_apply] |
| rw [proposalFiber,referenceFiber] |
| rw [fiber_lintegral checks (fun i ↦ (μ i).compProd (κ i)) s o |
| (fun i ↦ (x i,z i)) (fun p ↦ f (fun i ↦ (p i).1)), |
| fiber_lintegral reference μ s o x |
| (fun p ↦ conditionalLikelihood reference checks μ κ s o (Function.updateFinset x s p) z*f p)] |
| have hw := conditionalDensity_weighted checks μ κ s o x z F hF |
| have he : (∫⁻p,f (fun i ↦ (p i).1)*checkWeight checks (s∪o) |
| (Function.updateFinset (fun i ↦ (x i,z i)) s p) |
| ∂Measure.pi (fun i : s ↦ (μ i).compProd (κ i)))= |
| lmarginal μ s (fun y ↦ F y*conditionalDensity checks κ s o y z) x := by |
| rw [hw] |
| apply lintegral_congr |
| intro p |
| congr 1 |
| change f (fun i ↦ (p i).1)=f (fun i : s ↦ (Function.updateFinset (fun i ↦ (x i,z i)) s p i).1) |
| congr 1 |
| funext i |
| simp [Function.updateFinset,i.property] |
| rw [he] |
| have hp (p : ∀i : s,X i) : |
| (conditionalLikelihood reference checks μ κ s o (Function.updateFinset x s p) z*f p)* |
| checkWeight reference (s∪o) (Function.updateFinset x s p)= |
| (Z₀/Y₀)*(F (Function.updateFinset x s p)* |
| conditionalDensity checks κ s o (Function.updateFinset x s p) z) := by |
| rw [conditionalLikelihood_update checks reference μ κ,hu] |
| have hh := conditionalDensity_reference_support reference checks κ s o hincl |
| (Function.updateFinset x s p) z |
| calc |
| _ = (Z₀/Y₀)*(f p*(checkWeight reference (s∪o) (Function.updateFinset x s p)* |
| conditionalDensity checks κ s o (Function.updateFinset x s p) z)) := by ac_rfl |
| _ = _ := by rw [hh] |
| simp only [hp] |
| have hprod : Measurable (fun p : ∀i:s,X i ↦ |
| F (Function.updateFinset x s p)*conditionalDensity checks κ s o (Function.updateFinset x s p) z) := |
| (hF.mul (measurable_conditionalDensity checks κ s o z)).comp |
| (measurable_updateFinset (x:=x) (s:=s)) |
| rw [lintegral_const_mul _ hprod] |
| change Y₀⁻¹ * _ = Z₀⁻¹ *((Z₀/Y₀)*_) |
| rw [ENNReal.div_eq_inv_mul] |
| calc |
| _ = (Z₀⁻¹*Z₀)*(Y₀⁻¹*lmarginal μ s (fun y ↦ F y*conditionalDensity checks κ s o y z) x) := by rw [hZm,one_mul] |
| _ = _ := by ac_rfl |
|
|
| end HardSphere.Comparison |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
|
|
