| import Mathlib |
| import OAI.Probability.SKBarriers.Gaussian.GaussianZeroMass |
|
|
| 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 |
| namespace SK.Analytic |
| section StepCalculus |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
|
|
| def gaussianStepLaw (m : ℝ) (f : E × ℝ → ℝ) (x : E) : Measure ℝ := |
| (gaussianReal 0 1).tilted (fun y => m*f (x,y)) |
|
|
| theorem BoundedDerivs.exp_integrable {f : E × ℝ → ℝ} (hf : BoundedDerivs f) |
| (m : ℝ) (x : E) : Integrable (fun y => Real.exp (m*f (x,y))) (gaussianReal 0 1) := by |
| obtain ⟨hf,C,D,hC,hD,hb,hbb⟩ := hf |
| exact (exp_mul_growth_of_fderiv_bound f (hf.differentiable (by norm_num)) C hC hb m).integrable_gaussian_section (Real.continuous_exp.comp (continuous_const.mul hf.continuous)) x |
|
|
| theorem gaussianStepLaw_probability {f : E × ℝ → ℝ} (hf : BoundedDerivs f) |
| (m : ℝ) (x : E) : IsProbabilityMeasure (gaussianStepLaw m f x) := |
| isProbabilityMeasure_tilted (hf.exp_integrable m x) |
|
|
| theorem gaussianTransition_density {f : E × ℝ → ℝ} (hf : BoundedDerivs f) |
| (m : ℝ) (x : E) (y : ℝ) : |
| Real.exp (m*(f (x,y)-gaussianStep m f x)) = |
| Real.exp (m*f (x,y)) / (∫ z, Real.exp (m*f (x,z)) ∂gaussianReal 0 1) := by |
| by_cases hm : m = 0 |
| · subst m |
| simp |
| · have hpos := integral_exp_pos (hf.exp_integrable m x) |
| rw [gaussianStep,ite_eq_right hm,positiveGaussianLogStep] |
| have he : m*(f (x,y)-Real.log (∫ z, Real.exp (m*f (x,z)) ∂gaussianReal 0 1)/m) = |
| m*f (x,y)-Real.log (∫ z, Real.exp (m*f (x,z)) ∂gaussianReal 0 1) := by |
| rw [mul_sub, mul_div_cancel₀ _ hm] |
| rw [he,Real.exp_sub,Real.exp_log hpos] |
|
|
| theorem gaussianTransition_integral_one {f : E × ℝ → ℝ} (hf : BoundedDerivs f) |
| (m : ℝ) (x : E) : |
| (∫ y, Real.exp (m*(f (x,y)-gaussianStep m f x)) ∂gaussianReal 0 1) = 1 := by |
| simp_rw [gaussianTransition_density hf] |
| rw [integral_div,div_self (integral_exp_pos (hf.exp_integrable m x)).ne'] |
| |
| theorem gaussianStepLaw_integral {F : Type} [NormedAddCommGroup F] [NormedSpace ℝ F] |
| {f : E × ℝ → ℝ} (hf : BoundedDerivs f) (m : ℝ) (x : E) (g : ℝ → F) : |
| (∫ y, g y ∂gaussianStepLaw m f x) = |
| ∫ y, Real.exp (m*(f (x,y)-gaussianStep m f x)) • g y ∂gaussianReal 0 1 := by |
| rw [gaussianStepLaw,integral_tilted] |
| simp_rw [gaussianTransition_density hf] |
| |
| omit [NormedAddCommGroup E] [NormedSpace ℝ E] in |
| theorem gaussianStepLaw_zero (f : E × ℝ → ℝ) (x : E) : |
| gaussianStepLaw 0 f x = gaussianReal 0 1 := by |
| simp [gaussianStepLaw] |
| |
| theorem BoundedDerivs.gaussianStep {f : E × ℝ → ℝ} (hf : BoundedDerivs f) (m : ℝ) : |
| BoundedDerivs (gaussianStep m f) := by |
| unfold SK.Analytic.gaussianStep |
| split |
| · exact hf.gaussian_integral |
| · exact hf.positiveGaussianLogStep m |
| |
| theorem fderiv_gaussianStep {f : E × ℝ → ℝ} (hf : BoundedDerivs f) |
| (m : ℝ) (x : E) : |
| fderiv ℝ (gaussianStep m f) x = |
| ∫ y, leftRestrict (fderiv ℝ f (x,y)) ∂gaussianStepLaw m f x := by |
| classical |
| obtain ⟨hfd,C,D,hC,hD,hb,hbb⟩ := hf |
| have hd := hfd.differentiable (by norm_num) |
| have hg := exp_mul_growth_of_fderiv_bound f hd C hC hb m |
| have hmf : BoundedDerivs (fun z => m*f z) := |
| BoundedDerivs.const_mul ⟨hfd,C,D,hC,hD,hb,hbb⟩ m |
| have hmfb : ∀ z, ‖fderiv ℝ (fun z => m*f z) z‖ ≤ |m| * C := by |
| intro z |
| rw [((hd z).hasFDerivAt.const_mul m).fderiv, norm_smul, Real.norm_eq_abs] |
| exact mul_le_mul_of_nonneg_left (hb z) (abs_nonneg m) |
| have hg₁ := fderiv_exp_growth _ (hmf.1.differentiable (by norm_num)) hg |
| (mul_nonneg (abs_nonneg m) hC) hmfb |
| by_cases hm : m = 0 |
| · subst m |
| rw [gaussianStepLaw_zero] |
| have he : SK.Analytic.gaussianStep 0 f = (fun x => ∫ y, f (x,y) ∂gaussianReal 0 1) := by |
| funext x |
| simp only [SK.Analytic.gaussianStep,ite_true] |
| rw [he] |
| exact fderiv_gaussian_integral f (hfd.of_le (by norm_num)) |
| (growth_of_fderiv_bound f hd hC hb) |
| (HasExpGrowth.of_bounded hC hb) x |
| · let Z : E → ℝ := fun x => ∫ y, Real.exp (m*f (x,y)) ∂gaussianReal 0 1 |
| have hZ := hmf.exp_smooth_integral |
| have hp : Z x ≠ 0 := (integral_exp_pos |
| (BoundedDerivs.exp_integrable ⟨hfd,C,D,hC,hD,hb,hbb⟩ m x)).ne' |
| have he : gaussianStep m f = fun x => Real.log (Z x)/m := by |
| funext x |
| simp only [gaussianStep,ite_eq_right hm,positiveGaussianLogStep,Z] |
| rw [he] |
| have hder := ((hZ.differentiable (by norm_num) x).hasFDerivAt.log hp).mul_const m⁻¹ |
| simp only [div_eq_mul_inv] at ⊢ |
| rw [hder.fderiv] |
| have hi := fderiv_gaussian_integral (fun z => Real.exp (m*f z)) |
| ((contDiff_const.mul hfd).exp.of_le (by norm_num)) hg hg₁ x |
| change (m⁻¹ • ((Z x)⁻¹ • fderiv ℝ Z x)) = _ |
| rw [hi,← integral_smul,← integral_smul,gaussianStepLaw,integral_tilted] |
| apply integral_congr_ae |
| filter_upwards [] with y |
| have hdexp := ((hd (x,y)).hasFDerivAt.const_mul m).exp.fderiv |
| rw [hdexp] |
| ext v |
| simp only [smul_apply,smul_eq_mul,leftRestrict, |
| ContinuousLinearMap.compL_apply,ContinuousLinearMap.flip_apply, |
| ContinuousLinearMap.comp_apply,ContinuousLinearMap.inl_apply] |
| dsimp only [Z] at hp ⊢ |
| field_simp [hm,hp] |
|
|
| end StepCalculus |
| end SK.Analytic |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
|
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| end OAI |
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