| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.AuxiliaryChecks |
|
|
| 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 NormalizerFacts |
| variable {ι : Type*} [DecidableEq ι] {X : ι → Type*} |
| [∀ i, MeasurableSpace (X i)] {C : Type*} [Fintype C] |
| (checks : C → LocalCheck X) |
| omit [Fintype C] in |
| lemma partialNormalizer_le_one {μ : ∀ i, Measure (X i)} [∀ i, IsProbabilityMeasure (μ i)] |
| (s o : Finset ι) (x : ∀ i, X i) : partialNormalizer checks μ s o x ≤ 1 := by |
| calc |
| _ ≤ ∫⁻ _, (1 : ℝ≥0∞) ∂Measure.pi (fun i : s ↦ μ i) := |
| lintegral_mono fun _ ↦ checkWeight_le_one checks _ _ |
| _ = 1 := by simp |
|
|
| lemma measurable_partialNormalizer {μ : ∀ i, Measure (X i)} [∀ i, SigmaFinite (μ i)] |
| (s o : Finset ι) : Measurable (partialNormalizer checks μ s o) := |
| (measurable_checkWeight checks (s ∪ o)).lmarginal μ |
|
|
| omit [Fintype C] in |
| lemma UniformInsertionOn.mono {μ : ∀ i, Measure (X i)} {a : ℝ≥0∞} |
| {s t : Finset ι} (h : UniformInsertionOn checks μ a t) (hst : s ⊆ t) : |
| UniformInsertionOn checks μ a s := fun i hi ↦ h i (hst hi) |
|
|
| lemma checkNormalizer_pos_on {μ : ∀ i, Measure (X i)} [∀ i, SigmaFinite (μ i)] |
| {a : ℝ≥0∞} (ha : 0 < a) (s : Finset ι) |
| (hcompat : UniformInsertionOn checks μ a s) (x : ∀ i, X i) : |
| 0 < checkNormalizer checks μ s x := by |
| have hp := partialNormalizer_pos checks ha hcompat (Finset.disjoint_empty_right s) x |
| (show x ∈ retainedEvent checks ∅ from fun c hc ↦ |
| False.elim ((checks c).nonempty.ne_empty (Finset.subset_empty.mp hc))) |
| simpa only [partialNormalizer, checkNormalizer, Finset.union_empty] using hp |
|
|
| lemma checkNormalizer_union_lower_on {μ : ∀ i, Measure (X i)} [∀ i, SigmaFinite (μ i)] |
| {a : ℝ≥0∞} {t : Finset ι} (hcompat : UniformInsertionOn checks μ a t) |
| (s : Finset ι) (hst : Disjoint s t) (x : ∀ i, X i) : |
| a ^ t.card * checkNormalizer checks μ s x ≤ checkNormalizer checks μ (s ∪ t) x := by |
| have h := partialNormalizer_union_lower checks hcompat s t ∅ (Finset.Subset.refl t) |
| (by simpa using hst) x |
| simpa only [partialNormalizer, checkNormalizer, Finset.union_empty] using h |
|
|
| end NormalizerFacts |
|
|
| lemma isProbabilityMeasure_compProd_of_mass {X Z : Type*} [MeasurableSpace X] |
| [MeasurableSpace Z] (μ : Measure X) [IsFiniteMeasure μ] |
| (κ : Kernel X Z) [IsFiniteKernel κ] (hm : ∫⁻ x, κ x Set.univ ∂μ = 1) : |
| IsProbabilityMeasure (μ.compProd κ) where |
| measure_univ := by simpa only [Measure.compProd_apply MeasurableSet.univ, |
| Set.preimage_univ] using hm |
|
|
| section ConditionalOverlap |
| variable {ι : Type*} [DecidableEq ι] {X Z W : ι → Type*} |
| [∀ i, MeasurableSpace (X i)] [∀ i, MeasurableSpace (Z i)] |
| [∀ i, MeasurableSpace (W i)] |
| {C D : Type*} [Fintype C] [Fintype D] |
| (checks : C → LocalCheck (fun i ↦ X i × Z i)) |
| (other : D → LocalCheck (fun i ↦ X i × W i)) |
| (κ : ∀ i, Kernel (X i) (Z i)) [∀ i, IsFiniteKernel (κ i)] |
| (η : ∀ i, Kernel (X i) (W i)) [∀ i, IsFiniteKernel (η i)] |
|
|
| lemma conditionalDensity_overlap_bound |
| {μ : ∀ i, Measure (X i)} [∀ i, IsProbabilityMeasure (μ i)] |
| (fs ft cs ct os ot : Finset ι) |
| (hff : Disjoint fs ft) (hfn : Disjoint (fs ∪ ft) (cs ∪ ct)) |
| {G : ℝ≥0∞} |
| (hκb : ∀ i x, κ i x Set.univ ≤ G) |
| (hκm : ∀ i, ∫⁻ x, κ i x Set.univ ∂μ i = 1) |
| (hηm : ∀ i, ∫⁻ x, η i x Set.univ ∂μ i = 1) |
| (x : ∀ i, X i) (z : ∀ i, Z i) (w : ∀ i, W i) : |
| lmarginal μ ((fs ∪ ft) ∪ (cs ∪ ct)) |
| (fun y ↦ conditionalDensity checks κ (fs ∪ cs) os y z * |
| conditionalDensity other η (ft ∪ ct) ot y w) x ≤ |
| (checkNormalizer checks (fun i ↦ (μ i).compProd (κ i)) fs (fun i ↦ (x i,z i)) * |
| checkNormalizer other (fun i ↦ (μ i).compProd (η i)) ft (fun i ↦ (x i,w i))) * |
| G ^ (cs ∩ ct).card := by |
| apply le_trans (lmarginal_mono (fun y ↦ |
| mul_le_mul' (conditionalDensity_le_auxiliaryDensity checks κ _ _ y z) |
| (conditionalDensity_le_auxiliaryDensity other η _ _ y w)) x) |
| exact auxiliaryDensity_overlap_bound checks other κ η fs ft cs ct hff hfn hκb hκm hηm x z w |
|
|
| end ConditionalOverlap |
|
|
| lemma finite_normalizer_cancellation {p q zs zt ys yt : ℝ≥0∞} |
| (hp0 : p ≠ 0) (hpT : p ≠ ⊤) (hq0 : q ≠ 0) (hqT : q ≠ ⊤) |
| (hzs0 : zs ≠ 0) (hzsT : zs ≠ ⊤) (hzt0 : zt ≠ 0) (hztT : zt ≠ ⊤) |
| (hys0 : ys ≠ 0) (hysT : ys ≠ ⊤) (hyt0 : yt ≠ 0) (hytT : yt ≠ ⊤) |
| (b : ℝ≥0∞) : |
| ((zs / (p * ys)) * (zt / (q * yt)) * ((ys * yt) * b)) / |
| ((p * q) * (zs * zt)) = b / ((p * q) ^ 2) := by |
| have hpq0 : p * q ≠ 0 := mul_ne_zero hp0 hq0 |
| have hpqT : p * q ≠ ⊤ := ENNReal.mul_ne_top hpT hqT |
| simp only [div_eq_mul_inv, |
| ENNReal.mul_inv (Or.inl hp0) (Or.inl hpT), |
| ENNReal.mul_inv (Or.inl hq0) (Or.inl hqT), |
| ENNReal.mul_inv (Or.inl hpq0) (Or.inl hpqT), |
| ENNReal.mul_inv (Or.inl hzs0) (Or.inl hzsT), ENNReal.inv_pow] |
| calc |
| _ = (zs * zs⁻¹) * (zt * zt⁻¹) * (ys * ys⁻¹) * (yt * yt⁻¹) * |
| (b * (p⁻¹ * q⁻¹) ^ 2) := by simp only [pow_two]; ac_rfl |
| _ = _ := by rw [ENNReal.mul_inv_cancel hzs0 hzsT, |
| ENNReal.mul_inv_cancel hzt0 hztT, ENNReal.mul_inv_cancel hys0 hysT, |
| ENNReal.mul_inv_cancel hyt0 hytT]; simp |
|
|
| section ConditionalLower |
| variable {ι : Type*} [DecidableEq ι] {X : ι → Type*} |
| [∀ i, MeasurableSpace (X i)] {C : Type*} [Fintype C] |
| (checks : C → LocalCheck X) |
|
|
| lemma partialNormalizer_ge_far {μ : ∀ i, Measure (X i)} [∀ i, SigmaFinite (μ i)] |
| {a : ℝ≥0∞} {s t o : Finset ι} |
| (hcompat : UniformInsertionOn checks μ a (s ∪ t)) |
| (hto : Disjoint (s ∪ o) t) (hso : Disjoint s o) |
| (hsep : SeparatedChecks checks s o) (x : ∀ i, X i) |
| (hx : x ∈ retainedEvent checks o) : |
| a ^ t.card * checkNormalizer checks μ s x ≤ |
| partialNormalizer checks μ (s ∪ t) o x := by |
| have h := partialNormalizer_union_lower checks hcompat s t o Finset.subset_union_right hto x |
| rwa [partialNormalizer_far_eq checks hso hsep x, checkWeight, |
| Set.indicator_of_mem hx, mul_one] at h |
|
|
| end ConditionalLower |
|
|
| section CancellationEstimate |
| variable {ι : Type*} [DecidableEq ι] {X Z W : ι → Type*} |
| [∀ i, MeasurableSpace (X i)] [∀ i, MeasurableSpace (Z i)] |
| [∀ i, MeasurableSpace (W i)] |
| {R C D : Type*} [Fintype R] [Fintype C] [Fintype D] |
| (reference : R → LocalCheck X) |
| (checks : C → LocalCheck (fun i ↦ X i × Z i)) |
| (other : D → LocalCheck (fun i ↦ X i × W i)) |
| (μ : ∀ i, Measure (X i)) [∀ i, IsProbabilityMeasure (μ i)] |
| (κ : ∀ i, Kernel (X i) (Z i)) [∀ i, IsFiniteKernel (κ i)] |
| (η : ∀ i, Kernel (X i) (W i)) [∀ i, IsFiniteKernel (η i)] |
|
|
| def conditionalLikelihood (s o : Finset ι) (x : ∀ i, X i) (z : ∀ i, Z i) : ℝ≥0∞ := |
| (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 x z |
|
|
| lemma measurable_conditionalLikelihood (s o : Finset ι) (z : ∀ i, Z i) : |
| Measurable (fun x ↦ conditionalLikelihood reference checks μ κ s o x z) := by |
| apply Measurable.mul |
| · apply Measurable.div (measurable_partialNormalizer reference s o) |
| exact (measurable_partialNormalizer checks s o).comp |
| (Measurable.of_eval fun i ↦ (measurable_pi_apply i).prodMk measurable_const) |
| · exact measurable_conditionalDensity checks κ s o z |
|
|
| theorem conditional_cancellation |
| (fs ft cs ct os ot : Finset ι) |
| (hff : Disjoint fs ft) (hfn : Disjoint (fs ∪ ft) (cs ∪ ct)) |
| (hos : Disjoint (fs ∪ cs) os) (hot : Disjoint (ft ∪ ct) ot) |
| (hosU : (fs ∪ cs) ∪ os = (fs ∪ ft) ∪ (cs ∪ ct)) |
| (hotU : (ft ∪ ct) ∪ ot = (fs ∪ ft) ∪ (cs ∪ ct)) |
| {a G : ℝ≥0∞} (ha : 0 < a) (ha1 : a ≤ 1) |
| (hκb : ∀ i x, κ i x Set.univ ≤ G) |
| (hκm : ∀ i, ∫⁻ x, κ i x Set.univ ∂μ i = 1) |
| (hηm : ∀ i, ∫⁻ x, η i x Set.univ ∂μ i = 1) |
| (hRef : UniformInsertionOn reference μ a ((fs ∪ ft) ∪ (cs ∪ ct))) |
| (hS : UniformInsertionOn checks (fun i ↦ (μ i).compProd (κ i)) a (fs ∪ cs)) |
| (hT : UniformInsertionOn other (fun i ↦ (μ i).compProd (η i)) a (ft ∪ ct)) |
| (hsepRef : SeparatedChecks reference fs ft) |
| (hsepS : SeparatedChecks checks fs os) (hsepT : SeparatedChecks other ft ot) |
| (hOutsideS : ∀ (x : ∀ i, X i) (z : ∀ i, Z i), x ∈ retainedEvent reference os → |
| (fun i ↦ (x i,z i)) ∈ retainedEvent checks os) |
| (hOutsideT : ∀ (x : ∀ i, X i) (w : ∀ i, W i), x ∈ retainedEvent reference ot → |
| (fun i ↦ (x i,w i)) ∈ retainedEvent other ot) |
| (x : ∀ i, X i) (z : ∀ i, Z i) (w : ∀ i, W i) : |
| lmarginal μ ((fs ∪ ft) ∪ (cs ∪ ct)) |
| (fun y ↦ checkWeight reference ((fs ∪ ft) ∪ (cs ∪ ct)) y * |
| conditionalLikelihood reference checks μ κ (fs ∪ cs) os y z * |
| conditionalLikelihood reference other μ η (ft ∪ ct) ot y w) x / |
| checkNormalizer reference μ ((fs ∪ ft) ∪ (cs ∪ ct)) x ≤ |
| G ^ (cs ∩ ct).card / a ^ (2 * (cs.card + ct.card)) := by |
| let U := (fs ∪ ft) ∪ (cs ∪ ct) |
| let ν i := (μ i).compProd (κ i) |
| let ξ i := (μ i).compProd (η i) |
| have : ∀ i, IsProbabilityMeasure (ν i) := |
| fun i ↦ isProbabilityMeasure_compProd_of_mass (μ i) (κ i) (hκm i) |
| have : ∀ i, IsProbabilityMeasure (ξ i) := |
| fun i ↦ isProbabilityMeasure_compProd_of_mass (μ i) (η i) (hηm i) |
| let zs := checkNormalizer reference μ fs x |
| let zt := checkNormalizer reference μ ft x |
| let ys := checkNormalizer checks ν fs (fun i ↦ (x i,z i)) |
| let yt := checkNormalizer other ξ ft (fun i ↦ (x i,w i)) |
| have hfsU : fs ⊆ U := Finset.subset_union_left.trans Finset.subset_union_left |
| have hftU : ft ⊆ U := Finset.subset_union_right.trans Finset.subset_union_left |
| have hNearU : cs ∪ ct ⊆ U := Finset.subset_union_right |
| have hzs : 0 < zs := checkNormalizer_pos_on reference ha fs |
| (UniformInsertionOn.mono reference hRef hfsU) x |
| have hzt : 0 < zt := checkNormalizer_pos_on reference ha ft |
| (UniformInsertionOn.mono reference hRef hftU) x |
| have hys : 0 < ys := checkNormalizer_pos_on checks ha fs |
| (UniformInsertionOn.mono checks hS Finset.subset_union_left) _ |
| have hyt : 0 < yt := checkNormalizer_pos_on other ha ft |
| (UniformInsertionOn.mono other hT Finset.subset_union_left) _ |
| have hzsT : zs ≠ ⊤ := ne_top_of_le_ne_top ENNReal.one_ne_top (checkNormalizer_le_one reference fs x) |
| have hztT : zt ≠ ⊤ := ne_top_of_le_ne_top ENNReal.one_ne_top (checkNormalizer_le_one reference ft x) |
| have hysT : ys ≠ ⊤ := ne_top_of_le_ne_top ENNReal.one_ne_top (checkNormalizer_le_one checks fs _) |
| have hytT : yt ≠ ⊤ := ne_top_of_le_ne_top ENNReal.one_ne_top (checkNormalizer_le_one other ft _) |
| have haT : a ≠ ⊤ := ne_top_of_le_ne_top ENNReal.one_ne_top ha1 |
| have hZU : a ^ (cs.card + ct.card) * (zs * zt) ≤ checkNormalizer reference μ U x := by |
| calc |
| _ ≤ a ^ (cs ∪ ct).card * (zs * zt) := |
| mul_le_mul_left (pow_le_pow_right_of_le_one' ha1 (Finset.card_union_le cs ct)) _ |
| _ = a ^ (cs ∪ ct).card * checkNormalizer reference μ (fs ∪ ft) x := by |
| rw [checkNormalizer_union_eq_mul reference hff hsepRef] |
| _ ≤ _ := checkNormalizer_union_lower_on reference |
| (UniformInsertionOn.mono reference hRef hNearU) (fs ∪ ft) hfn x |
| have hfs : Disjoint fs cs := hfn.mono Finset.subset_union_left Finset.subset_union_left |
| have hft : Disjoint ft ct := hfn.mono Finset.subset_union_right Finset.subset_union_right |
| have hfsos : Disjoint fs os := hos.mono_left Finset.subset_union_left |
| have hftot : Disjoint ft ot := hot.mono_left Finset.subset_union_left |
| have hfcso : Disjoint (fs ∪ os) cs := Finset.disjoint_union_left.mpr |
| ⟨hfs, (hos.mono_left Finset.subset_union_right).symm⟩ |
| have hftcto : Disjoint (ft ∪ ot) ct := Finset.disjoint_union_left.mpr |
| ⟨hft, (hot.mono_left Finset.subset_union_right).symm⟩ |
| let rs := zs / (a ^ cs.card * ys) |
| let rt := zt / (a ^ ct.card * yt) |
| have hratioS (y : ∀ i, X i) (hy : y ∈ retainedEvent reference U) : |
| partialNormalizer reference μ (fs ∪ cs) os y / |
| partialNormalizer checks ν (fs ∪ cs) os (fun i ↦ (y i,z i)) ≤ rs := by |
| have hyo : y ∈ retainedEvent reference os := fun c hc ↦ hy c |
| (hc.trans (by change os ⊆ (fs ∪ ft) ∪ (cs ∪ ct); rw [← hosU]; exact Finset.subset_union_right)) |
| have hl := partialNormalizer_ge_far checks hS hfcso hfsos hsepS |
| (fun i ↦ (y i,z i)) (hOutsideS y z hyo) |
| rw [checkNormalizer_const checks fs _ (fun i ↦ (x i,z i))] at hl |
| have hu := partialNormalizer_le_far reference (μ := μ) (o := os) hfs y |
| rw [checkNormalizer_const reference fs y x] at hu |
| exact ENNReal.div_le_div hu hl |
| have hratioT (y : ∀ i, X i) (hy : y ∈ retainedEvent reference U) : |
| partialNormalizer reference μ (ft ∪ ct) ot y / |
| partialNormalizer other ξ (ft ∪ ct) ot (fun i ↦ (y i,w i)) ≤ rt := by |
| have hyo : y ∈ retainedEvent reference ot := fun c hc ↦ hy c |
| (hc.trans (by change ot ⊆ (fs ∪ ft) ∪ (cs ∪ ct); rw [← hotU]; exact Finset.subset_union_right)) |
| have hl := partialNormalizer_ge_far other hT hftcto hftot hsepT |
| (fun i ↦ (y i,w i)) (hOutsideT y w hyo) |
| rw [checkNormalizer_const other ft _ (fun i ↦ (x i,w i))] at hl |
| have hu := partialNormalizer_le_far reference (μ := μ) (o := ot) hft y |
| rw [checkNormalizer_const reference ft y x] at hu |
| exact ENNReal.div_le_div hu hl |
| let H (y : ∀ i, X i) := conditionalDensity checks κ (fs ∪ cs) os y z * |
| conditionalDensity other η (ft ∪ ct) ot y w |
| have hH : Measurable H := (measurable_conditionalDensity checks κ _ _ z).mul |
| (measurable_conditionalDensity other η _ _ w) |
| have hpoint (y : ∀ i, X i) : |
| checkWeight reference U y * conditionalLikelihood reference checks μ κ (fs ∪ cs) os y z * |
| conditionalLikelihood reference other μ η (ft ∪ ct) ot y w ≤ (rs * rt) * H y := by |
| classical |
| by_cases hy : y ∈ retainedEvent reference U |
| · simp only [checkWeight, Set.indicator_of_mem hy, one_mul, conditionalLikelihood] |
| calc |
| _ ≤ (rs * conditionalDensity checks κ (fs ∪ cs) os y z) * |
| (rt * conditionalDensity other η (ft ∪ ct) ot y w) := |
| mul_le_mul' (mul_le_mul_left (hratioS y hy) _) (mul_le_mul_left (hratioT y hy) _) |
| _ = _ := by dsimp [H]; ac_rfl |
| · simp [checkWeight, hy] |
| have hIntegral : lmarginal μ U |
| (fun y ↦ checkWeight reference U y * |
| conditionalLikelihood reference checks μ κ (fs ∪ cs) os y z * |
| conditionalLikelihood reference other μ η (ft ∪ ct) ot y w) x ≤ |
| (rs * rt) * ((ys * yt) * G ^ (cs ∩ ct).card) := by |
| calc |
| _ ≤ lmarginal μ U (fun y ↦ (rs * rt) * H y) x := lmarginal_mono hpoint x |
| _ = (rs * rt) * lmarginal μ U H x := |
| lintegral_const_mul _ (hH.comp (measurable_updateFinset (x := x) (s := U))) |
| _ ≤ _ := mul_le_mul_right |
| (conditionalDensity_overlap_bound checks other κ η fs ft cs ct os ot hff hfn |
| hκb hκm hηm x z w) _ |
| calc |
| _ ≤ ((rs * rt) * ((ys * yt) * G ^ (cs ∩ ct).card)) / |
| (a ^ (cs.card + ct.card) * (zs * zt)) := ENNReal.div_le_div hIntegral hZU |
| _ = G ^ (cs ∩ ct).card / ((a ^ cs.card * a ^ ct.card) ^ 2) := by |
| rw [pow_add] |
| exact finite_normalizer_cancellation (ne_of_gt (ENNReal.pow_pos ha _)) |
| (ENNReal.pow_ne_top haT) (ne_of_gt (ENNReal.pow_pos ha _)) (ENNReal.pow_ne_top haT) |
| (ne_of_gt hzs) hzsT (ne_of_gt hzt) hztT (ne_of_gt hys) hysT (ne_of_gt hyt) hytT _ |
| _ = _ := by rw [← pow_add, ← pow_mul, Nat.mul_comm 2] |
|
|
| end CancellationEstimate |
|
|
| section LikelihoodIdentity |
| 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)] |
|
|
| omit [Fintype R] [∀ i, IsProbabilityMeasure (μ i)] in |
| lemma partialNormalizer_updateFinset (s o : Finset ι) (x : ∀ i, X i) (u : ∀ i : s, X i) : |
| partialNormalizer reference μ s o (Function.updateFinset x s u) = |
| partialNormalizer reference μ s o x := by |
| apply lmarginal_congr |
| intro i hi |
| simp [Function.updateFinset, hi] |
|
|
| omit [Fintype R] in |
| lemma partialNormalizer_le_outside {s o : Finset ι} (hso : Disjoint s o) (x : ∀ i, X i) : |
| partialNormalizer reference μ s o x ≤ checkWeight reference o x := by |
| calc |
| _ ≤ lmarginal μ s (checkWeight reference o) x := |
| lmarginal_mono (checkWeight_antitone reference Finset.subset_union_right) x |
| _ = _ := lmarginal_of_disjoint_depends |
| (fun _ _ h ↦ checkWeight_depends reference o h) hso.symm x |
|
|
| omit [Fintype R] [Fintype C] [∀ i, IsFiniteKernel (κ i)] in |
|
|
| lemma conditionalDensity_reference_support (s o : Finset ι) |
| (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) : |
| checkWeight reference (s ∪ o) x * conditionalDensity checks κ s o x z = |
| conditionalDensity checks κ s o x z := by |
| classical |
| by_cases hx : x ∈ retainedEvent reference (s ∪ o) |
| · simp [checkWeight, hx] |
| · have hz : conditionalDensity checks κ s o x z = 0 := by |
| apply le_antisymm _ zero_le |
| calc |
| _ ≤ lmarginal (fun i ↦ κ i (x i)) s (fun _ ↦ checkWeight reference (s ∪ o) x) z := |
| lmarginal_mono (hincl x) z |
| _ = 0 := by simp [lmarginal, checkWeight, hx] |
| simp [hz] |
|
|
| omit [Fintype R] in |
|
|
| theorem conditionalLikelihood_marginal (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) |
| (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) : |
| lmarginal μ s (fun y ↦ checkWeight reference (s ∪ o) y * |
| conditionalLikelihood reference checks μ κ s o y z) x = |
| partialNormalizer reference μ s o x := by |
| let ν i := (μ i).compProd (κ i) |
| have : ∀ i, IsProbabilityMeasure (ν i) := |
| fun i ↦ isProbabilityMeasure_compProd_of_mass (μ i) (κ i) (hκm i) |
| let Z₀ := partialNormalizer reference μ s o x |
| let Y₀ := partialNormalizer checks ν s o (fun i ↦ (x i,z i)) |
| have hYu (u : ∀ i : s, X i) : |
| partialNormalizer checks ν s o (fun i ↦ (Function.updateFinset x s u i,z i)) = Y₀ := by |
| apply lmarginal_congr |
| intro i hi |
| simp [Function.updateFinset, hi] |
| have hRatio : ∀ (u : ∀ i : s, X i), |
| partialNormalizer reference μ s o (Function.updateFinset x s u) / |
| partialNormalizer checks ν s o (fun i ↦ (Function.updateFinset x s u i,z i)) = Z₀ / Y₀ := by |
| intro u |
| rw [partialNormalizer_updateFinset, hYu] |
| have hm := measurable_conditionalDensity checks κ s o z |
| calc |
| _ = lmarginal μ s |
| (fun y ↦ (partialNormalizer reference μ s o y / |
| partialNormalizer checks ν s o (fun i ↦ (y i,z i))) * |
| conditionalDensity checks κ s o y z) x := by |
| apply lintegral_congr |
| intro u |
| change checkWeight reference (s ∪ o) (Function.updateFinset x s u) * |
| conditionalLikelihood reference checks μ κ s o (Function.updateFinset x s u) z = _ |
| dsimp only [conditionalLikelihood] |
| rw [mul_left_comm, conditionalDensity_reference_support reference checks κ s o hincl] |
| _ = (Z₀ / Y₀) * lmarginal μ s (fun y ↦ conditionalDensity checks κ s o y z) x := by |
| change (∫⁻ u, (partialNormalizer reference μ s o (Function.updateFinset x s u) / |
| partialNormalizer checks ν s o (fun i ↦ (Function.updateFinset x s u i,z i))) * |
| conditionalDensity checks κ s o (Function.updateFinset x s u) z |
| ∂Measure.pi (fun i : s ↦ μ i)) = _ |
| simp_rw [hRatio] |
| exact lintegral_const_mul _ (hm.comp (measurable_updateFinset (x := x) (s := s))) |
| _ = (Z₀ / Y₀) * Y₀ := by rw [conditionalDensity_integral] |
| _ = Z₀ := by |
| classical |
| by_cases hx : x ∈ retainedEvent reference o |
| · have hypos : 0 < Y₀ := partialNormalizer_pos checks ha hcompat hso _ (hOutside x z hx) |
| have hyT : Y₀ ≠ ⊤ := ne_top_of_le_ne_top ENNReal.one_ne_top |
| (partialNormalizer_le_one checks s o _) |
| exact ENNReal.div_mul_cancel (ne_of_gt hypos) hyT |
| · have hz0 : Z₀ = 0 := le_antisymm (by |
| simpa [checkWeight, hx] using partialNormalizer_le_outside reference μ hso x) zero_le |
| simp [hz0] |
|
|
| end LikelihoodIdentity |
|
|
| 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)] |
|
|
| def referenceMeasure (s : Finset ι) (x : ∀ i, X i) : Measure (∀ i : s, X i) := |
| (Measure.pi (fun i : s ↦ μ i)).withDensity |
| (fun u ↦ checkWeight reference s (Function.updateFinset x s u) / |
| checkNormalizer reference μ s x) |
|
|
| omit [∀ i, IsProbabilityMeasure (μ i)] in |
| lemma referenceMeasure_mass (s : Finset ι) (x : ∀ i, X i) : |
| referenceMeasure reference μ s x Set.univ = |
| checkNormalizer reference μ s x / checkNormalizer reference μ s x := by |
| rw [referenceMeasure, withDensity_apply _ MeasurableSet.univ, |
| Measure.restrict_univ] |
| simp only [div_eq_mul_inv] |
| exact lintegral_mul_const _ |
| ((measurable_checkWeight reference s).comp measurable_updateFinset) |
|
|
| lemma referenceMeasure_probability {a : ℝ≥0∞} (ha : 0 < a) (s : Finset ι) |
| (hcompat : UniformInsertionOn reference μ a s) (x : ∀ i, X i) : |
| IsProbabilityMeasure (referenceMeasure reference μ s x) := by |
| refine ⟨?_⟩ |
| rw [referenceMeasure_mass] |
| exact ENNReal.div_self (ne_of_gt (checkNormalizer_pos_on reference ha s hcompat x)) |
| (ne_top_of_le_ne_top ENNReal.one_ne_top (checkNormalizer_le_one reference (μ := μ) s x)) |
|
|
| end HardSphere.Comparison |
| end |
|
|
| end OAI |
|
|
