| import Mathlib |
| import OAI.Probability.SKBarriers.Calculus.ParameterGrowth |
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| namespace OAI |
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| section |
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| 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 |
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| end |
| end |
|
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| end |
|
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| end OAI |
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