| import Mathlib |
| import OAI.Probability.SKRatio.Quantization.BinPartition |
| import OAI.Probability.SKRatio.Variational.FieldFunctional |
|
|
| namespace OAI |
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|
| noncomputable section |
| open scoped NNReal ENNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Real |
| namespace SKRatio.Bins |
| open Planted Scalar |
| variable {α : Type*} [Fintype α] [MeasurableSpace α] |
| [MeasurableSingletonClass α] {μ : Measure ℝ} [IsProbabilityMeasure μ] |
|
|
| lemma integral_finite_partition_prod {σ : ℝ → α} (hσ : Measurable σ) (F : α → α → ℝ) : |
| (∫ z : ℝ×ℝ, F (σ z.1) (σ z.2) ∂μ.prod μ)= |
| ∑ a, ∑ d, binMass μ σ a*binMass μ σ d*F a d := by |
| have hi : Integrable (fun z : ℝ×ℝ => F (σ z.1) (σ z.2)) (μ.prod μ) := |
| (memLp_finite_comp ((hσ.comp measurable_fst).prodMk (hσ.comp measurable_snd)) |
| (fun ad : α×α => F ad.1 ad.2)).integrable (by norm_num : (1:ENNReal) ≤ 2) |
| rw [integral_prod _ hi] |
| simp_rw [integral_finite_partition (μ := μ) hσ (F (σ _))] |
| rw [integral_finite_partition (μ := μ) hσ (fun a => ∑ d, binMass μ σ d*F a d)] |
| simp only [Finset.mul_sum,mul_assoc] |
|
|
| lemma fieldRadicand_pullback {σ : ℝ → α} (hσ : Measurable σ) |
| (hm : ∀ a, 0 < binMass μ σ a) (h : α → ℝ) (z : Parameters α) (x : ℝ) : |
| fieldRadicand μ (fun x => h (σ x)) (pullback μ σ (paramB z)) |
| (pullback μ σ (paramR z)) x= |
| finiteRadicalSq (massRoot μ σ) (fun a => v (h a)) z (σ x) := by |
| rw [fieldRadicand_expansion (memLp_pullback hσ _) (memLp_pullback hσ _) |
| (measurable_finite_comp hσ h)] |
| rw [integral_pullback_energy hσ hm (paramB z) (paramR z) |
| (fun a => (v (h (σ x))+v (h a))^2)] |
| simp_rw [mul_comm (v (h (σ x))+v (h (σ _))) (pullback μ σ (paramB z) _)] |
| rw [integral_pullback hσ hm (paramB z) (fun a => v (h (σ x))+v (h a))] |
| rfl |
|
|
| lemma fieldDiagonal_partition {σ : ℝ → α} (hσ : Measurable σ) (h : α → ℝ) (x : ℝ) : |
| fieldDiagonal μ (fun x => h (σ x)) x= |
| ∑ d, massRoot μ σ d^2*(v (h d)^2/(w (h (σ x))+w (h d))) := by |
| simp_rw [massRoot_sq] |
| exact integral_finite_partition hσ (fun d => v (h d)^2/(w (h (σ x))+w (h d))) |
|
|
| lemma field_diagonal_pullback {σ : ℝ → α} (hσ : Measurable σ) |
| (hm : ∀ a, 0 < binMass μ σ a) (β : ℝ) (h : α → ℝ) (z : Parameters α) : |
| (∫ x, (β^2*fieldDiagonal μ (fun x => h (σ x)) x)* |
| (pullback μ σ (paramB z) x^2+pullback μ σ (paramR z) x^2) ∂μ)= |
| β^2*(∑ a, ∑ d, massRoot μ σ d^2*(v (h d)^2/(w (h a)+w (h d)))* |
| ((paramB z a)^2+(paramR z a)^2)) := by |
| simp_rw [fieldDiagonal_partition hσ h,mul_assoc,integral_const_mul] |
| rw [integral_pullback_energy hσ hm (paramB z) (paramR z) |
| (fun a => ∑ d, massRoot μ σ d^2*(v (h d)^2/(w (h a)+w (h d))))] |
| simp only [Finset.sum_mul, mul_assoc] |
|
|
| lemma field_kernel_pullback {σ : ℝ → α} (hσ : Measurable σ) |
| (hm : ∀ a, 0 < binMass μ σ a) (h : α → ℝ) (B : α → ℝ) : |
| (∫ z : ℝ×ℝ, pullback μ σ B z.1*kernel (h (σ z.1)) (h (σ z.2))* |
| pullback μ σ B z.2 ∂μ.prod μ)= |
| ∑ a, ∑ d, massRoot μ σ a*B a*kernel (h a) (h d)*(massRoot μ σ d*B d) := by |
| change (∫ z : ℝ×ℝ, (fun a d => (B a/massRoot μ σ a)*kernel (h a) (h d)* |
| (B d/massRoot μ σ d)) (σ z.1) (σ z.2) ∂μ.prod μ)=_ |
| rw [integral_finite_partition_prod (μ := μ) hσ (fun a d => (B a/massRoot μ σ a)*kernel (h a) (h d)*(B d/massRoot μ σ d))] |
| apply Finset.sum_congr rfl |
| intro a _ |
| apply Finset.sum_congr rfl |
| intro d _ |
| rw [←massRoot_sq σ a,←massRoot_sq σ d] |
| field_simp [(massRoot_pos hm a).ne',(massRoot_pos hm d).ne'] |
|
|
| theorem fieldForm_pullback {σ : ℝ → α} (hσ : Measurable σ) |
| (hm : ∀ a, 0 < binMass μ σ a) (β : ℝ) (h : α → ℝ) (z : Parameters α) : |
| fieldForm μ β (fun x => h (σ x)) (pullback μ σ (paramB z)) |
| (pullback μ σ (paramR z))=finiteV β (massRoot μ σ) h z := by |
| unfold fieldForm |
| rw [field_diagonal_pullback hσ hm β h z,field_kernel_pullback hσ hm] |
| simp_rw [fieldRadicand_pullback hσ hm h z] |
| rw [integral_pullback hσ hm (paramB z) h, |
| integral_pullback hσ hm (paramB z) (fun a => v (h a)), |
| integral_pullback hσ hm (paramB z) (fun a => h a*v (h a)), |
| integral_pullback hσ hm (paramB z) (fun a => m (h a)*v (h a)), |
| integral_pullback hσ hm (paramR z) (fun a => sqrt (finiteRadicalSq (massRoot μ σ) (fun a => v (h a)) z a))] |
| have hmean : (∫ x, pullback μ σ (paramB z) x ∂μ)=moment (massRoot μ σ) (paramB z) (fun _ => 1) := by |
| simpa only [mul_one] using integral_pullback hσ hm (paramB z) (fun _ => 1) |
| rw [hmean] |
| unfold finiteV finiteA finiteGaussian moment |
| ring |
|
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| end SKRatio.Bins |
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| end |
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| end OAI |
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