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b317718 | 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 | import OAI.Probability.GaussianPropeller.ProbabilityBounds
namespace OAI
universe uE
open MeasureTheory ProbabilityTheory
open scoped ENNReal
open scoped RealInnerProductSpace
open scoped RealInnerProductSpace
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal RealInnerProductSpace
open Filter
open scoped Topology
open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology
open scoped RealInnerProductSpace
open Set Filter
open scoped Topology RealInnerProductSpace
open scoped NNReal
open Set Filter
open scoped Topology RealInnerProductSpace NNReal
open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology RealInnerProductSpace
open MeasureTheory Set Filter
open scoped Topology BigOperators
open MeasureTheory ProbabilityTheory Set Filter
open scoped RealInnerProductSpace Topology
namespace GaussianPropeller.Analytic
theorem integral_pos_mul_exp {b : ℝ} (hb : 0 < b) :
∫ r : ℝ in Ioi 0, r * Real.exp (-b * r ^ 2) = (2 * b)⁻¹ := by
have hb' : b ≠ 0 := ne_of_gt hb
have hd (x : ℝ) : HasDerivAt (fun x => -(2 * b)⁻¹ * Real.exp (-b * x ^ 2))
(x * Real.exp (-b * x ^ 2)) x := by
apply (((hasDerivAt_pow 2 x).const_mul (-b)).exp.const_mul (-(2 * b)⁻¹)).congr_deriv
field_simp [hb']
ring
have ht : Tendsto (fun y : ℝ => -(2 * b)⁻¹ * Real.exp (-b * y ^ 2))
atTop (𝓝 (-(2 * b)⁻¹ * 0)) := by
apply Tendsto.const_mul
exact Real.tendsto_exp_atBot.comp
((tendsto_pow_atTop two_ne_zero).const_mul_atTop_of_neg (neg_lt_zero.mpr hb))
convert integral_Ioi_of_hasDerivAt_of_tendsto' (fun x _ => hd x)
(integrable_mul_exp_neg_mul_sq hb).integrableOn ht using 1
simp
theorem integral_abs_mul_exp {b : ℝ} (hb : 0 < b) :
∫ r : ℝ, |r| * Real.exp (-b * r ^ 2) = b⁻¹ := by
let f : ℝ → ℝ := fun r => |r| * Real.exp (-b * r ^ 2)
have hf : Integrable f := by
simpa only [f, abs_mul, abs_of_pos (Real.exp_pos _)] using
(integrable_mul_exp_neg_mul_sq hb).abs
have hpos : ∫ r in Ioi 0, f r = (2 * b)⁻¹ := by
rw [← integral_pos_mul_exp hb]
exact setIntegral_congr_fun measurableSet_Ioi (fun r hr => by
simp only [f, abs_of_pos (show (0 : ℝ) < r from hr)])
have hneg : ∫ r in Iic 0, f r = ∫ r in Ioi 0, f r := by
calc
∫ r in Iic 0, f r = ∫ r in Ioi 0, f (-r) := by
simpa [f] using (integral_comp_neg_Ioi 0 f).symm
_ = ∫ r in Ioi 0, f r := by simp only [f, abs_neg, neg_sq]
change (∫ r, f r) = _
rw [← integral_add_compl measurableSet_Ioi hf, compl_Ioi, hneg, hpos]
field_simp
ring
theorem integral_abs_gaussianReal_std :
∫ x : ℝ, |x| ∂gaussianReal 0 1 = 2 / Real.sqrt (2 * Real.pi) := by
rw [integral_gaussianReal_eq_integral_smul (by norm_num : (1 : NNReal) ≠ 0)]
have hfun : (fun x : ℝ => gaussianPDFReal 0 1 x • |x|) =
(fun x : ℝ => (Real.sqrt (2 * Real.pi))⁻¹ *
(|x| * Real.exp (-(1 / 2 : ℝ) * x ^ 2))) := by
ext x
simp only [gaussianPDFReal, NNReal.coe_one, sub_zero, mul_one, smul_eq_mul]
rw [show -(x ^ 2) / (2 : ℝ) = -(1 / 2 : ℝ) * x ^ 2 by ring]
ring
rw [hfun, integral_const_mul, integral_abs_mul_exp (by norm_num : (0 : ℝ) < 1/2)]
norm_num
ring
theorem integral_abs_dual_stdGaussian {E : Type uE} [NormedAddCommGroup E]
[InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
(L : StrongDual ℝ E) :
∫ x, |L x| ∂stdGaussian E = ‖L‖ * (2 / Real.sqrt (2 * Real.pi)) := by
have hm : (stdGaussian E).map L = (gaussianReal 0 1).map (‖L‖ * ·) := by
rw [IsGaussian.map_eq_gaussianReal, integral_strongDual_stdGaussian,
variance_dual_stdGaussian, gaussianReal_map_const_mul]
simp only [mul_zero, mul_one, Real.toNNReal_of_nonneg (sq_nonneg ‖L‖)]
rw [← integral_map L.continuous.aemeasurable (by fun_prop :
AEStronglyMeasurable (fun x : ℝ => |x|) ((stdGaussian E).map L)), hm,
integral_map (by fun_prop) (by fun_prop)]
simp_rw [abs_mul, abs_of_nonneg (norm_nonneg L)]
rw [integral_const_mul, integral_abs_gaussianReal_std]
theorem integral_abs_inner_stdGaussian {E : Type uE} [NormedAddCommGroup E]
[InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
(v : E) :
∫ x, |⟪v, x⟫| ∂stdGaussian E = ‖v‖ * (2 / Real.sqrt (2 * Real.pi)) := by
simpa only [InnerProductSpace.toDual_apply_apply, LinearIsometryEquiv.norm_map] using
integral_abs_dual_stdGaussian (InnerProductSpace.toDual ℝ E v)
end GaussianPropeller.Analytic
end OAI
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