File size: 5,769 Bytes
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 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 | import Mathlib
import OAI.Probability.SKBarriers.Calculus.ParameterGrowth
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter
namespace SK.Analytic
section ParameterLinear
variable {P E : Type} [NormedAddCommGroup P] [NormedAddCommGroup E]
theorem ParamLinearGrowth.const_mul {f : P × E → ℝ} (hf : ParamLinearGrowth f) (m : ℝ) :
ParamLinearGrowth (fun z => m*f z) := by
intro R
obtain ⟨C,hC,hf⟩ := hf R
refine ⟨|m| *C,mul_nonneg (abs_nonneg _) hC,?_⟩
intro p e hp
rw [abs_mul]
exact (mul_le_mul_of_nonneg_left (hf p e hp) (abs_nonneg _)).trans_eq (mul_assoc _ _ _).symm
theorem ParamLinearGrowth.gaussian_integral {f : P × (E × ℝ) → ℝ}
(hf : ParamLinearGrowth f) :
ParamLinearGrowth (fun z : P × E => ∫ y, f (z.1,(z.2,y)) ∂gaussianReal 0 1) := by
intro R
obtain ⟨C,hC,hf⟩ := hf R
let B := ∫ y : ℝ, Real.exp |y| ∂gaussianReal 0 1
have hB : 0 ≤ B := integral_nonneg (fun _ => (Real.exp_pos _).le)
have hI : Integrable (fun y : ℝ => Real.exp |y|) (gaussianReal 0 1) := by
simpa only [one_mul] using integrable_exp_mul_abs_gaussian 1
refine ⟨C*B,mul_nonneg hC hB,?_⟩
intro p e hp
have hh y : |f (p,(e,y))| ≤ C*(1+‖e‖)*Real.exp |y| := by
apply (hf p (e,y) hp).trans
rw [Prod.norm_def,Real.norm_eq_abs]
have hmax := max_le_add_of_nonneg (norm_nonneg e) (abs_nonneg y)
have hex := Real.add_one_le_exp |y|
have hprod := mul_nonneg (norm_nonneg e) (abs_nonneg y)
have H : 1+max ‖e‖ |y| ≤ (1+‖e‖)*Real.exp |y| := by
calc
_ ≤ (1+‖e‖)*(1+|y|) := by nlinarith
_ ≤ _ := mul_le_mul_of_nonneg_left (by linarith) (by positivity)
exact (mul_le_mul_of_nonneg_left H hC).trans_eq (mul_assoc _ _ _).symm
calc
_ = ‖∫ y, f (p,(e,y)) ∂gaussianReal 0 1‖ := (Real.norm_eq_abs _).symm
_ ≤ ∫ y, ‖f (p,(e,y))‖ ∂gaussianReal 0 1 := norm_integral_le_integral_norm _
_ ≤ ∫ y, C*(1+‖e‖)*Real.exp |y| ∂gaussianReal 0 1 :=
integral_mono_of_nonneg (ae_of_all _ (fun _ => norm_nonneg _)) (hI.const_mul _)
(ae_of_all _ hh)
_ = _ := by rw [integral_const_mul]; dsimp [B]; ring
end ParameterLinear
theorem gaussian_exp_integral_bounds (g : ℝ → ℝ) (hg : Continuous g) {A C : ℝ}
(_hC : 0 ≤ C) (hb : ∀ y, |g y| ≤ A+C*|y|) :
let Z := ∫ y, Real.exp (g y) ∂gaussianReal 0 1
Real.exp (-A)*(∫ y : ℝ, Real.exp (-C*|y|) ∂gaussianReal 0 1) ≤ Z ∧
Z ≤ Real.exp A*(∫ y : ℝ, Real.exp (C*|y|) ∂gaussianReal 0 1) := by
have hupper y : Real.exp (g y) ≤ Real.exp A*Real.exp (C*|y|) := by
rw [← Real.exp_add]
exact Real.exp_le_exp.2 ((le_abs_self _).trans (hb y))
have hlower y : Real.exp (-A)*Real.exp (-C*|y|) ≤ Real.exp (g y) := by
rw [← Real.exp_add]
apply Real.exp_le_exp.2
have hh := (abs_le.1 (hb y)).1
linarith
have hI : Integrable (fun y => Real.exp (g y)) (gaussianReal 0 1) := by
apply ((integrable_exp_mul_abs_gaussian C).const_mul (Real.exp A)).mono' (Real.continuous_exp.comp hg).aestronglyMeasurable
exact ae_of_all _ (fun y => by simpa only [Function.comp_apply,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] using hupper y)
dsimp only
constructor
· rw [← integral_const_mul]
exact integral_mono ((integrable_exp_mul_abs_gaussian (-C)).const_mul _) hI hlower
· rw [← integral_const_mul]
exact integral_mono hI ((integrable_exp_mul_abs_gaussian C).const_mul _) hupper
section LogParameterLinear
variable {P E : Type} [NormedAddCommGroup P] [NormedAddCommGroup E]
theorem ParamLinearGrowth.log_gaussian_exp {f : P × (E × ℝ) → ℝ}
(hf : ParamLinearGrowth f) (hc : Continuous f) :
ParamLinearGrowth (fun z : P × E => Real.log (∫ y, Real.exp (f (z.1,(z.2,y))) ∂gaussianReal 0 1)) := by
intro R
obtain ⟨C,hC,hf⟩ := hf R
let Bp := ∫ y : ℝ, Real.exp (C*|y|) ∂gaussianReal 0 1
let Bm := ∫ y : ℝ, Real.exp (-C*|y|) ∂gaussianReal 0 1
have hp : 0 < Bp := integral_exp_pos (integrable_exp_mul_abs_gaussian C)
have hm : 0 < Bm := integral_exp_pos (integrable_exp_mul_abs_gaussian (-C))
let D := C+|Real.log Bp|+|Real.log Bm|
refine ⟨D,by dsimp [D]; positivity,?_⟩
intro p e he
let A := C*(1+‖e‖)
have hb y : |f (p,(e,y))| ≤ A+C*|y| := by
apply (hf p (e,y) he).trans
rw [Prod.norm_def,Real.norm_eq_abs]
have hh := max_le_add_of_nonneg (norm_nonneg e) (abs_nonneg y)
dsimp [A]
nlinarith
have H := gaussian_exp_integral_bounds (fun y => f (p,(e,y)))
(hc.comp (continuous_const.prodMk (continuous_const.prodMk continuous_id))) hC hb
let Z := ∫ y, Real.exp (f (p,(e,y))) ∂gaussianReal 0 1
have hz : 0 < Z := lt_of_lt_of_le (mul_pos (Real.exp_pos _) hm) H.1
have hu := Real.log_le_log hz H.2
have hl := Real.log_le_log (mul_pos (Real.exp_pos _) hm) H.1
change Real.log Z ≤ Real.log (Real.exp A*Bp) at hu
change Real.log (Real.exp (-A)*Bm) ≤ Real.log Z at hl
rw [Real.log_mul (Real.exp_pos _).ne' hp.ne',Real.log_exp] at hu
rw [Real.log_mul (Real.exp_pos _).ne' hm.ne',Real.log_exp] at hl
change |Real.log Z| ≤ D*(1+‖e‖)
apply abs_le.2
constructor
· have hh := le_abs_self (Real.log Bm)
have hneg := neg_abs_le (Real.log Bm)
have hextra := mul_nonneg (add_nonneg (abs_nonneg (Real.log Bp)) (abs_nonneg (Real.log Bm))) (norm_nonneg e)
dsimp [A,D] at *
linarith [abs_nonneg (Real.log Bp)]
· have hh := le_abs_self (Real.log Bp)
have hextra := mul_nonneg (add_nonneg (abs_nonneg (Real.log Bp)) (abs_nonneg (Real.log Bm))) (norm_nonneg e)
dsimp [A,D] at *
linarith [abs_nonneg (Real.log Bm)]
end LogParameterLinear
end SK.Analytic
end
end
end
end OAI
|