| 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 |
| |
