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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 | import Mathlib
import OAI.Probability.SKBarriers.Calculus.ParameterDerivatives
import OAI.Probability.SKBarriers.Calculus.ParameterInverse
import OAI.Probability.SKBarriers.Calculus.ParameterSmooth
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter
namespace SK.Analytic
section ParameterRecursion
variable {P E : Type} [NormedAddCommGroup P] [NormedSpace ℝ P]
[NormedAddCommGroup E] [NormedSpace ℝ E]
theorem ParamRegular.log_gaussian_exp {f : P × (E × ℝ) → ℝ} (hf : ParamRegular f) :
ParamRegular (fun z : P × E => Real.log
(∫ y, Real.exp (f (z.1,(z.2,y))) ∂gaussianReal 0 1)) := by
have hg : ParamExpGrowth (fun z => Real.exp (f z)) := by
simpa only [one_mul] using hf.2.1.exp_mul 1
have h₁ := param_fderiv_exp_growth f id (hf.1.differentiable (by norm_num)) hg hf.2.2.1
have h₂ := param_second_exp_growth f id hf.1 hg hf.2.2.1 hf.2.2.2
let Z := fun z : P × E => ∫ y, Real.exp (f (z.1,(z.2,y))) ∂gaussianReal 0 1
have hZ := param_gaussian_smooth (fun z => Real.exp (f z)) hf.1.exp hg h₁ h₂
have hp (z : P × E) : Z z ≠ 0 := by
apply (integral_exp_pos _).ne'
apply hg.weakLocallyDominated.integrable_section
exact Real.continuous_exp.comp (hf.1.continuous.comp
(continuous_fst.fst.prodMk (continuous_fst.snd.prodMk continuous_snd)))
have hi : ParamExpGrowth (fun z => (Z z)⁻¹) := hf.2.1.inv_gaussian_exp hf.1.continuous
exact ⟨hZ.1.log hp,hf.2.1.log_gaussian_exp hf.1.continuous,
param_fderiv_log_growth Z id (hZ.1.differentiable (by norm_num)) hp hi hZ.2.1,
param_second_log_growth Z id hZ.1 hp hi hZ.2.1 hZ.2.2⟩
theorem ParamRegular.gaussianStep {f : P × (E × ℝ) → ℝ} (hf : ParamRegular f) (m : ℝ) :
ParamRegular (gaussianStep m (fun z : (P × E) × ℝ => f (z.1.1,(z.1.2,z.2)))) := by
change ParamRegular (fun z : P × E => if m = 0 then
∫ y, f (z.1,(z.2,y)) ∂gaussianReal 0 1 else
Real.log (∫ y, Real.exp (m*f (z.1,(z.2,y))) ∂gaussianReal 0 1) / m)
by_cases hm : m = 0
· subst m
simpa only [ite_true] using hf.gaussian_integral
· have H := ((hf.const_mul m).log_gaussian_exp).const_mul m⁻¹
simpa only [ite_eq_right hm,div_eq_mul_inv,mul_comm] using H
end ParameterRecursion
end SK.Analytic
end
end
end
end OAI
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