File size: 5,563 Bytes
f61e547 | 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 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367
theorem hasDerivAt_gaussianPDFReal (m : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (x : ℝ) :
HasDerivAt (gaussianPDFReal m v)
(-(x - m) / (v : ℝ) * gaussianPDFReal m v x) x := by
have hv' : (v : ℝ) ≠ 0 := by exact_mod_cast hv
have h := (((((hasDerivAt_id x).sub_const m).pow 2).neg.div_const
(2 * (v : ℝ))).exp).const_mul ((Real.sqrt (2 * Real.pi * v))⁻¹)
convert! h using 1
simp only [gaussianPDFReal, Pi.neg_apply, Pi.pow_apply, id_eq,
Nat.cast_ofNat, Nat.reduceSub, pow_one, mul_one]
field_simp
theorem integrable_gaussian_iff_density {m : ℝ} {v : ℝ≥0} (hv : v ≠ 0)
{f : ℝ → ℝ} :
Integrable f (gaussianReal m v) ↔
Integrable (fun x => gaussianPDFReal m v x * f x) := by
rw [gaussianReal_of_var_ne_zero _ hv]
simpa only [toReal_gaussianPDF, smul_eq_mul] using
(integrable_withDensity_iff_integrable_smul'
(measurable_gaussianPDF m v)
(Filter.Eventually.of_forall (fun _ => gaussianPDF_lt_top)) (g := f))
theorem gaussian_integral_centered_mul_eq_variance_mul_deriv
(m : ℝ) (v : ℝ≥0) {f f' : ℝ → ℝ}
(hf : ∀ x, HasDerivAt f (f' x) x)
(hfi : Integrable f (gaussianReal m v))
(hf'i : Integrable f' (gaussianReal m v))
(hxfi : Integrable (fun x => (x - m) * f x) (gaussianReal m v)) :
(∫ x, (x - m) * f x ∂gaussianReal m v) =
(v : ℝ) * ∫ x, f' x ∂gaussianReal m v := by
by_cases hv : v = 0
· subst v
simp
have hv' : (v : ℝ) ≠ 0 := by exact_mod_cast hv
have huf := (integrable_gaussian_iff_density hv).mp hfi
have huf' := (integrable_gaussian_iff_density hv).mp hf'i
have hxuf := (integrable_gaussian_iff_density hv).mp hxfi
have hfirst : Integrable (fun x => f x *
(-(x - m) / (v : ℝ) * gaussianPDFReal m v x)) := by
convert hxuf.const_mul (-(v : ℝ)⁻¹) using 1
ext x
ring
have hsecond : Integrable (fun x => f' x * gaussianPDFReal m v x) := by
simpa only [mul_comm] using huf'
have hproduct : Integrable (fun x => f x * gaussianPDFReal m v x) := by
simpa only [mul_comm] using huf
have hibp := integral_mul_deriv_eq_deriv_mul_of_integrable
(u := f) (v := gaussianPDFReal m v)
(u' := f') (v' := fun x => -(x - m) / (v : ℝ) * gaussianPDFReal m v x)
(fun x _ => hf x) (fun x _ => hasDerivAt_gaussianPDFReal m hv x)
hfirst hsecond hproduct
have hfun : (fun x => f x * (-(x - m) / (v : ℝ) * gaussianPDFReal m v x)) =
(fun x => -(v : ℝ)⁻¹ * (gaussianPDFReal m v x * ((x - m) * f x))) := by
funext x
ring
rw [hfun, integral_const_mul] at hibp
simp_rw [mul_comm (f' _) (gaussianPDFReal m v _)] at hibp
rw [integral_gaussianReal_eq_integral_smul hv,
integral_gaussianReal_eq_integral_smul hv]
simp only [smul_eq_mul]
have hmul := congrArg (fun z : ℝ => -(v : ℝ) * z) hibp
field_simp at hmul
simpa only [mul_assoc] using hmul
theorem gaussian_integral_centered_mul_of_bounded
(m : ℝ) (v : ℝ≥0) {f f' : ℝ → ℝ} {C : ℝ}
(hf : ∀ x, HasDerivAt f (f' x) x) (hbound : ∀ x, |f x| ≤ C)
(hf'i : Integrable f' (gaussianReal m v)) :
(∫ x, (x - m) * f x ∂gaussianReal m v) =
(v : ℝ) * ∫ x, f' x ∂gaussianReal m v := by
have hfm : AEStronglyMeasurable f (gaussianReal m v) :=
(continuous_iff_continuousAt.mpr (fun x => (hf x).continuousAt)).aestronglyMeasurable
have hb : ∀ᵐ x ∂gaussianReal m v, ‖f x‖ ≤ C :=
Filter.Eventually.of_forall (fun x => by simpa only [Real.norm_eq_abs] using hbound x)
have hfi := Integrable.of_bound hfm C hb
have hxi : Integrable (fun x : ℝ => x - m) (gaussianReal m v) :=
((memLp_id_gaussianReal 1).integrable (by norm_num)).sub (integrable_const m)
exact gaussian_integral_centered_mul_eq_variance_mul_deriv m v hf hfi hf'i
(hxi.mul_bdd hfm hb)
theorem gaussian_integral_centered_mul_of_bounded_deriv
(m : ℝ) (v : ℝ≥0) {f f' : ℝ → ℝ} {C D : ℝ}
(hf : ∀ x, HasDerivAt f (f' x) x)
(hbound : ∀ x, |f x| ≤ C) (hbound' : ∀ x, |f' x| ≤ D) :
(∫ x, (x - m) * f x ∂gaussianReal m v) =
(v : ℝ) * ∫ x, f' x ∂gaussianReal m v := by
apply gaussian_integral_centered_mul_of_bounded m v hf hbound
have heq : f' = deriv f := funext (fun x => (hf x).deriv.symm)
have hfm : AEStronglyMeasurable f' (gaussianReal m v) := by
rw [heq]
exact aestronglyMeasurable_deriv f _
exact Integrable.of_bound hfm D
(Filter.Eventually.of_forall (fun x => by simpa only [Real.norm_eq_abs] using hbound' x))
theorem gaussian_integral_mul_affine_of_bounded_deriv
(a b : ℝ) {f f' : ℝ → ℝ} {C D : ℝ}
(hf : ∀ x, HasDerivAt f (f' x) x)
(hbound : ∀ x, |f x| ≤ C) (hbound' : ∀ x, |f' x| ≤ D) :
(∫ z, z * f (a + b * z) ∂gaussianReal 0 1) =
b * ∫ z, f' (a + b * z) ∂gaussianReal 0 1 := by
have hderiv (z : ℝ) : HasDerivAt (fun z => f (a + b * z))
(b * f' (a + b * z)) z := by
simpa only [Function.comp_def, id_eq, mul_one, mul_comm] using (hf (a + b * z)).comp z
(((hasDerivAt_id z).const_mul b).const_add a)
have hb' (z : ℝ) : |b * f' (a + b * z)| ≤ |b| * D := by
rw [abs_mul]
exact mul_le_mul_of_nonneg_left (hbound' _) (abs_nonneg b)
have h := gaussian_integral_centered_mul_of_bounded_deriv 0 1 hderiv
(fun z => hbound (a + b * z)) hb'
simpa only [sub_zero, NNReal.coe_one, one_mul, integral_const_mul] using h
end Problem367
end
end OAI
|