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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 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 | import Mathlib
import OAI.Probability.SKBarriers.Calculus.ParameterAlgebra
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter
namespace SK.Analytic
section ParameterFiniteLog
variable {P E S : Type} [NormedAddCommGroup P] [NormedSpace ℝ P]
[NormedAddCommGroup E] [NormedSpace ℝ E] [Fintype S] [Nonempty S]
omit [NormedSpace ℝ P] [NormedSpace ℝ E] in
theorem param_inv_finite_exp_growth (f : S → P × E → ℝ)
(hf : ∀ s, ParamLinearGrowth (f s)) :
ParamExpGrowth (fun z => (∑ s, Real.exp (f s z))⁻¹) := by
classical
let s₀ : S := Classical.arbitrary S
apply ((hf s₀).exp_mul (-1)).of_norm_le zero_le_one
intro z
have hp : 0 < ∑ s, Real.exp (f s z) := Finset.sum_pos (fun _ _ => Real.exp_pos _) Finset.univ_nonempty
have hs : Real.exp (f s₀ z) ≤ ∑ s, Real.exp (f s z) :=
Finset.single_le_sum (fun s _ => (Real.exp_pos (f s z)).le) (Finset.mem_univ s₀)
rw [one_mul,Real.norm_eq_abs,Real.norm_eq_abs,abs_of_pos (inv_pos.2 hp),abs_of_pos (Real.exp_pos _),
neg_one_mul,Real.exp_neg]
exact (inv_le_inv₀ hp (Real.exp_pos _)).2 hs
omit [NormedSpace ℝ P] [NormedSpace ℝ E] in
theorem param_log_finite_exp_linear (f : S → P × E → ℝ)
(hf : ∀ s, ParamLinearGrowth (f s)) :
ParamLinearGrowth (fun z => Real.log (∑ s, Real.exp (f s z))) := by
classical
intro R
choose C hC hb using fun s => hf s R
let D := ∑ s, C s
let N : ℝ := Fintype.card S
have hD : 0 ≤ D := Finset.sum_nonneg (fun s _ => hC s)
have hN : 0 < N := by
dsimp [N]
exact_mod_cast Fintype.card_pos (α := S)
refine ⟨D+|Real.log N|,by positivity,?_⟩
intro p e hp
let A := D*(1+‖e‖)
let Z := ∑ s, Real.exp (f s (p,e))
have hz : 0 < Z := Finset.sum_pos (fun _ _ => Real.exp_pos _) Finset.univ_nonempty
have hh s : |f s (p,e)| ≤ A := by
apply (hb s p e hp).trans
apply mul_le_mul_of_nonneg_right _ (by positivity)
exact Finset.single_le_sum (fun t _ => hC t) (Finset.mem_univ s)
have hu' : Z ≤ N*Real.exp A := by
calc
Z ≤ ∑ _ : S, Real.exp A := Finset.sum_le_sum (fun s _ =>
Real.exp_le_exp.2 ((le_abs_self _).trans (hh s)))
_ = N*Real.exp A := by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,N]
let s₀ : S := Classical.arbitrary S
have hl' : Real.exp (-A) ≤ Z := by
calc
_ ≤ Real.exp (f s₀ (p,e)) := Real.exp_le_exp.2 (abs_le.1 (hh s₀)).1
_ ≤ Z := Finset.single_le_sum (fun s _ => (Real.exp_pos (f s (p,e))).le) (Finset.mem_univ s₀)
have hu := Real.log_le_log hz hu'
have hl := Real.log_le_log (Real.exp_pos _) hl'
rw [Real.log_mul hN.ne' (Real.exp_pos _).ne',Real.log_exp] at hu
rw [Real.log_exp] at hl
change |Real.log Z| ≤ (D+|Real.log N|)*(1+‖e‖)
apply abs_le.2
have hextra := mul_nonneg (abs_nonneg (Real.log N)) (norm_nonneg e)
dsimp [A] at hu hl
constructor
· linarith [abs_nonneg (Real.log N)]
· linarith [le_abs_self (Real.log N),abs_nonneg (Real.log N)]
theorem paramRegular_log_finite_exp (f : S → P × E → ℝ) (hf : ∀ s, ParamRegular (f s)) :
ParamRegular (fun z => Real.log (∑ s, Real.exp (f s z))) := by
classical
let Z := fun z : P × E => ∑ s, Real.exp (f s z)
have hZ : ParamSmooth Z := ParamSmooth.sum Finset.univ _ (fun s _ => (hf s).expSmooth)
have hp (z : P × E) : Z z ≠ 0 :=
(Finset.sum_pos (fun s (_ : s ∈ Finset.univ) => Real.exp_pos (f s z)) Finset.univ_nonempty).ne'
have hi : ParamExpGrowth (fun z => (Z z)⁻¹) := param_inv_finite_exp_growth f (fun s => (hf s).2.1)
exact ⟨hZ.1.log hp,param_log_finite_exp_linear f (fun s => (hf s).2.1),
param_fderiv_log_growth Z id (hZ.1.differentiable (by norm_num)) hp hi hZ.2.2.1,
param_second_log_growth Z id hZ.1 hp hi hZ.2.2.1 hZ.2.2.2⟩
end ParameterFiniteLog
end SK.Analytic
end
end
end
end OAI
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