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135052a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | import Mathlib
import OAI.Probability.SKBarriers.Gaussian.GaussianStepCalculus
import OAI.Probability.SKBarriers.Gaussian.LocalMarginal
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter
namespace SK.Analytic
section LocalStepDerivative
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
theorem fderiv_local_gaussianStep {f : E × ℝ → ℝ} (hf : ContDiff ℝ 1 f)
(hg : GaussianLocallyDominated f) (hg₁ : GaussianLocallyDominated (fderiv ℝ f))
(m : ℝ) (he : GaussianLocallyDominated (fun z => Real.exp (m*f z)))
(he₁ : GaussianLocallyDominated (fderiv ℝ (fun z => Real.exp (m*f z)))) (x : E) :
fderiv ℝ (gaussianStep m f) x =
∫ y, leftRestrict (fderiv ℝ f (x,y)) ∂gaussianStepLaw m f x := by
have hd := hf.differentiable (by norm_num)
by_cases hm : m = 0
· subst m
rw [gaussianStepLaw_zero]
have hfun : 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 [hfun]
exact fderiv_local_gaussian_integral f hf hg hg₁ x
· let Z : E → ℝ := fun x => ∫ y, Real.exp (m*f (x,y)) ∂gaussianReal 0 1
have hc : ContDiff ℝ 1 (fun z => Real.exp (m*f z)) := (contDiff_const.mul hf).exp
have hp : Z x ≠ 0 := (integral_exp_pos (he.integrable_section hc.continuous x)).ne'
have hi := hasFDerivAt_local_gaussian_integral (fun z => Real.exp (m*f z)) _
hc.continuous (leftRestrict.continuous.comp
((hc.fderiv_right (m := 0) (by norm_num)).continuous)) he (he₁.clm leftRestrict)
(leftRestrict_hasFDerivAt _ (hc.differentiable (by norm_num))) x
have hfun : gaussianStep m f = fun x => Real.log (Z x)/m := by
funext x
simp only [gaussianStep,ite_eq_right hm,positiveGaussianLogStep,Z]
rw [hfun]
have hder := (hi.log hp).mul_const m⁻¹
simp only [div_eq_mul_inv] at ⊢
rw [hder.fderiv]
rw [← 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
simp only [Function.comp_apply]
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 LocalStepDerivative
end SK.Analytic
end
end
end
end OAI
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