| import Mathlib |
| import OAI.Probability.SKRatio.Quantization.BinFunctional |
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| 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 {Ω α : Type*} [MeasurableSpace Ω] [Fintype α] [MeasurableSpace α] |
| [MeasurableSingletonClass α] {μ : Measure Ω} [IsProbabilityMeasure μ] |
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| def binMass (μ : Measure Ω) (σ : Ω → α) (a : α) : ℝ := μ.real {x | σ x=a} |
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| def massRoot (μ : Measure Ω) (σ : Ω → α) (a : α) : ℝ := sqrt (binMass μ σ a) |
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| def pullback (μ : Measure Ω) (σ : Ω → α) (B : α → ℝ) (x : Ω) : ℝ := |
| B (σ x)/massRoot μ σ (σ x) |
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| lemma integral_finite_partition {σ : Ω → α} (hσ : Measurable σ) (f : α → ℝ) : |
| (∫ x, f (σ x) ∂μ)=∑ a, binMass μ σ a*f a := by |
| rw [←integral_map hσ.aemeasurable (measurable_of_countable f).aestronglyMeasurable, |
| integral_fintype (by exact Integrable.of_finite)] |
| apply Finset.sum_congr rfl |
| intro a _ |
| rw [smul_eq_mul,map_measureReal_apply hσ (measurableSet_singleton a)] |
| rfl |
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| omit [Fintype α] [MeasurableSpace α] [MeasurableSingletonClass α] [IsProbabilityMeasure μ] in |
| lemma binMass_nonneg (σ : Ω → α) (a : α) : 0 ≤ binMass μ σ a := measureReal_nonneg |
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| omit [Fintype α] [MeasurableSpace α] [MeasurableSingletonClass α] [IsProbabilityMeasure μ] in |
| lemma massRoot_sq (σ : Ω → α) (a : α) : massRoot μ σ a^2=binMass μ σ a := |
| sq_sqrt (binMass_nonneg σ a) |
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| lemma sum_binMass {σ : Ω → α} (hσ : Measurable σ) : (∑ a, binMass μ σ a)=1 := by |
| have hh := integral_finite_partition (μ := μ) hσ (fun _ => (1:ℝ)) |
| simpa only [mul_one,integral_const,probReal_univ,smul_eq_mul,one_mul] using hh.symm |
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| lemma massRoot_unit {σ : Ω → α} (hσ : Measurable σ) : (∑ a, massRoot μ σ a^2)=1 := by |
| simp_rw [massRoot_sq] |
| exact sum_binMass hσ |
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| lemma measurable_finite_comp {σ : Ω → α} (hσ : Measurable σ) (f : α → ℝ) : |
| Measurable (fun x => f (σ x)) := (measurable_of_countable f).comp hσ |
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| lemma memLp_finite_comp {σ : Ω → α} (hσ : Measurable σ) (f : α → ℝ) : |
| MemLp (fun x => f (σ x)) 2 μ := by |
| apply MemLp.of_bound (measurable_finite_comp hσ f).aestronglyMeasurable (∑ a, |f a|) |
| exact ae_of_all _ (fun x => by |
| rw [norm_eq_abs] |
| exact Finset.single_le_sum (fun a _ => abs_nonneg (f a)) (Finset.mem_univ (σ x))) |
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| omit [IsProbabilityMeasure μ] in |
| lemma measurable_pullback {σ : Ω → α} (hσ : Measurable σ) (B : α → ℝ) : |
| Measurable (pullback μ σ B) := measurable_finite_comp hσ (fun a => B a / massRoot μ σ a) |
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| lemma memLp_pullback {σ : Ω → α} (hσ : Measurable σ) (B : α → ℝ) : |
| MemLp (pullback μ σ B) 2 μ := memLp_finite_comp hσ (fun a => B a / massRoot μ σ a) |
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| omit [Fintype α] [MeasurableSpace α] [MeasurableSingletonClass α] [IsProbabilityMeasure μ] in |
| lemma massRoot_pos {σ : Ω → α} (hm : ∀ a, 0<binMass μ σ a) (a : α) : 0 < massRoot μ σ a := |
| sqrt_pos.mpr (hm a) |
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| lemma integral_pullback {σ : Ω → α} (hσ : Measurable σ) (hm : ∀ a, 0<binMass μ σ a) |
| (B f : α → ℝ) : (∫ x, pullback μ σ B x*f (σ x) ∂μ)=moment (massRoot μ σ) B f := by |
| rw [show (fun x => pullback μ σ B x*f (σ x))= |
| (fun x => (fun a => B a/massRoot μ σ a*f a) (σ x)) from rfl, |
| integral_finite_partition (μ := μ) hσ (fun a => B a / massRoot μ σ a * f a)] |
| unfold moment |
| apply Finset.sum_congr rfl |
| intro a _ |
| have hs := massRoot_sq (μ := μ) σ a |
| have hn := (massRoot_pos hm a).ne' |
| rw [←hs] |
| field_simp |
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| lemma integral_pullback_energy {σ : Ω → α} (hσ : Measurable σ) |
| (hm : ∀ a, 0<binMass μ σ a) (B R f : α → ℝ) : |
| (∫ x, f (σ x)*(pullback μ σ B x^2+pullback μ σ R x^2) ∂μ)= |
| ∑ a, f a*(B a^2+R a^2) := by |
| rw [show (fun x => f (σ x)*(pullback μ σ B x^2+pullback μ σ R x^2))= |
| (fun x => (fun a => f a*((B a/massRoot μ σ a)^2+(R a/massRoot μ σ a)^2)) (σ x)) from rfl, |
| integral_finite_partition (μ := μ) hσ (fun a => f a*((B a/massRoot μ σ a)^2+(R a/massRoot μ σ a)^2))] |
| apply Finset.sum_congr rfl |
| intro a _ |
| rw [div_pow,div_pow,massRoot_sq] |
| field_simp [(hm a).ne'] |
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| lemma pullback_normalized [DecidableEq α] {σ : Ω → α} (hσ : Measurable σ) |
| (hm : ∀ a, 0<binMass μ σ a) (z : Parameters α) : |
| (∫ x, pullback μ σ (paramB z) x^2+pullback μ σ (paramR z) x^2 ∂μ)=1 := by |
| simpa only [one_mul] using (integral_pullback_energy hσ hm (paramB z) (paramR z) (fun _ => 1)).trans |
| (by simpa only [one_mul] using param_sum_sq z) |
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| end SKRatio.Bins |
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| end |
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| end OAI |
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