| import Mathlib |
| import OAI.Probability.SKBarriers.Gaussian.LogDerivatives |
| import OAI.Probability.SKBarriers.Gaussian.GaussianStepCalculus |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped Topology Interval |
| noncomputable section |
| open MeasureTheory Set |
| open scoped Interval |
| noncomputable section |
| open MeasureTheory Set Filter ProbabilityTheory |
| open scoped Topology |
| noncomputable section |
| open MeasureTheory Set Filter ProbabilityTheory |
| open scoped Topology NNReal |
| namespace SK.Analytic |
| section SecondStep |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
|
|
| theorem fderiv_fderiv_const_mul (f : E → ℝ) (hf : ContDiff ℝ 2 f) (c : ℝ) (x : E) : |
| fderiv ℝ (fderiv ℝ (fun z => c*f z)) x = c • fderiv ℝ (fderiv ℝ f) x := by |
| have he : fderiv ℝ (fun z => c*f z) = fun z => c • fderiv ℝ f z := by |
| funext z |
| exact ((hf.differentiable (by norm_num) z).hasFDerivAt.const_mul c).fderiv |
| rw [he] |
| exact (((hf.fderiv_right (m := 1) (by norm_num)).differentiable (by norm_num) x).hasFDerivAt.const_smul c).fderiv |
|
|
| theorem BoundedDerivs.exp_growths {f : E → ℝ} (hf : BoundedDerivs f) (m : ℝ) : |
| HasExpGrowth (fun x => Real.exp (m*f x)) ∧ |
| HasExpGrowth (fderiv ℝ (fun x => Real.exp (m*f x))) ∧ |
| HasExpGrowth (fderiv ℝ (fderiv ℝ (fun x => Real.exp (m*f x)))) := by |
| obtain ⟨hfd,C,D,hC,hD,hb,hbb⟩ := hf.const_mul m |
| have hg : HasExpGrowth (fun x => Real.exp (m*f x)) := by |
| simpa only [one_mul] using exp_mul_growth_of_fderiv_bound (fun x => m*f x) |
| (hfd.differentiable (by norm_num)) C hC hb 1 |
| exact ⟨hg,fderiv_exp_growth _ (hfd.differentiable (by norm_num)) hg hC hb, |
| fderiv_fderiv_exp_growth _ hfd hg hC hD hb hbb⟩ |
|
|
| theorem fderiv_fderiv_gaussianStep_nonzero {f : E × ℝ → ℝ} (hf : BoundedDerivs f) |
| {m : ℝ} (hm : m ≠ 0) (x : E) : |
| fderiv ℝ (fderiv ℝ (gaussianStep m f)) x = |
| m⁻¹ • (((∫ y, Real.exp (m*f (x,y)) ∂gaussianReal 0 1)⁻¹) • |
| (∫ y, secondLeftRestrict (fderiv ℝ (fderiv ℝ (fun z => Real.exp (m*f z))) (x,y)) |
| ∂gaussianReal 0 1) + |
| (-((∫ y, Real.exp (m*f (x,y)) ∂gaussianReal 0 1)^2)⁻¹ • |
| (∫ y, leftRestrict (fderiv ℝ (fun z => Real.exp (m*f z)) (x,y)) ∂gaussianReal 0 1)).smulRight |
| (∫ y, leftRestrict (fderiv ℝ (fun z => Real.exp (m*f z)) (x,y)) ∂gaussianReal 0 1)) := by |
| let Z : E → ℝ := fun x => ∫ y, Real.exp (m*f (x,y)) ∂gaussianReal 0 1 |
| have hZ := (hf.const_mul m).exp_smooth_integral |
| have hp (z : E) : Z z ≠ 0 := (integral_exp_pos (hf.exp_integrable m z)).ne' |
| have he : gaussianStep m f = fun z => m⁻¹*Real.log (Z z) := by |
| funext z |
| simp only [gaussianStep,ite_eq_right hm,positiveGaussianLogStep,Z,div_eq_mul_inv,mul_comm] |
| have hL : ContDiff ℝ 2 (fun z => Real.log (Z z)) := hZ.log hp |
| obtain ⟨hg,hg₁,hg₂⟩ := hf.exp_growths m |
| rw [he,fderiv_fderiv_const_mul _ hL,fderiv_fderiv_log _ hZ hp] |
| rw [fderiv_gaussian_integral _ ((contDiff_const.mul hf.1).exp.of_le (by norm_num)) hg hg₁] |
| rw [fderiv_fderiv_gaussian_integral _ (contDiff_const.mul hf.1).exp hg hg₁ hg₂] |
| |
| theorem fderiv_fderiv_gaussianStep_zero {f : E × ℝ → ℝ} (hf : BoundedDerivs f) (x : E) : |
| fderiv ℝ (fderiv ℝ (gaussianStep 0 f)) x = |
| ∫ y, secondLeftRestrict (fderiv ℝ (fderiv ℝ f) (x,y)) ∂gaussianReal 0 1 := by |
| have he : gaussianStep 0 f = fun x => ∫ y, f (x,y) ∂gaussianReal 0 1 := by |
| funext z |
| simp only [gaussianStep,ite_true] |
| rw [he] |
| obtain ⟨hfd,C,D,hC,hD,hb,hbb⟩ := hf |
| exact fderiv_fderiv_gaussian_integral f hfd |
| (growth_of_fderiv_bound f (hfd.differentiable (by norm_num)) hC hb) |
| (HasExpGrowth.of_bounded hC hb) (HasExpGrowth.of_bounded hD hbb) x |
| |
| theorem integrable_exp_smul_of_bounded {F : Type} [NormedAddCommGroup F] [NormedSpace ℝ F] |
| {f : E × ℝ → ℝ} (hf : BoundedDerivs f) (m : ℝ) (x : E) |
| (g : ℝ → F) (hg : Continuous g) {C : ℝ} (hb : ∀ y, ‖g y‖ ≤ C) : |
| Integrable (fun y => Real.exp (m*f (x,y)) • g y) (gaussianReal 0 1) := by |
| apply ((hf.exp_integrable m x).const_mul C).mono' |
| ((Real.continuous_exp.comp (continuous_const.mul |
| (hf.1.continuous.comp (continuous_const.prodMk continuous_id)))).smul hg).aestronglyMeasurable |
| filter_upwards [] with y |
| change ‖Real.exp (m*f (x,y)) • g y‖ ≤ C * Real.exp (m*f (x,y)) |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] |
| exact (mul_le_mul_of_nonneg_left (hb y) (Real.exp_pos _).le).trans_eq (mul_comm _ _) |
|
|
| end SecondStep |
| end SK.Analytic |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
