File size: 8,027 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 | import OAI.Probability.FreeIsing.Gaussian.IntegrationByParts
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367
section CoordinateSplit
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
theorem measurePreserving_funSplitAt (μ : ι → Measure ℝ)
[∀ j, SigmaFinite (μ j)] (i : ι) :
MeasurePreserving (Homeomorph.funSplitAt ℝ i) (Measure.pi μ)
((μ i).prod (Measure.pi fun j : {j // j ≠ i} => μ j)) := by
let e := (Homeomorph.funSplitAt ℝ i).toMeasurableEquiv.symm
apply MeasurePreserving.symm e
refine ⟨e.measurable, (Measure.pi_eq fun s hs => ?_).symm⟩
rw [e.map_apply]
have hpre : e ⁻¹' Set.pi Set.univ s =
s i ×ˢ Set.pi Set.univ (fun j : {j // j ≠ i} => s j) := by
ext x
simp only [Set.mem_preimage, Set.mem_pi, Set.mem_univ, forall_true_left,
Set.mem_prod]
constructor
· intro h
constructor
· simpa [e, Homeomorph.funSplitAt, Homeomorph.piSplitAt,
Equiv.piSplitAt] using h i
· intro j
simpa [e, Homeomorph.funSplitAt, Homeomorph.piSplitAt,
Equiv.piSplitAt, j.property] using h j
· rintro ⟨hi, hj⟩ j
by_cases hji : j = i
· subst j
simpa [e, Homeomorph.funSplitAt, Homeomorph.piSplitAt,
Equiv.piSplitAt] using hi
· simpa [e, Homeomorph.funSplitAt, Homeomorph.piSplitAt,
Equiv.piSplitAt, hji] using hj ⟨j, hji⟩
rw [hpre, Measure.prod_prod, Measure.pi_pi]
simpa using (Fintype.prod_eq_mul_prod_subtype_ne (fun j => μ j (s j)) i).symm
theorem gaussian_pi_integral_centered_mul_of_bounded_deriv
(m : ι → ℝ) (v : ι → ℝ≥0) (i : ι)
{f f' : (ι → ℝ) → ℝ} {C D : ℝ}
(hfc : Continuous f) (hf'c : Continuous f')
(hf : ∀ z t, HasDerivAt (fun r => f (Function.update z i r))
(f' (Function.update z i t)) t)
(hbound : ∀ z, |f z| ≤ C) (hbound' : ∀ z, |f' z| ≤ D) :
(∫ z, (z i - m i) * f z ∂Measure.pi (fun j => gaussianReal (m j) (v j))) =
(v i : ℝ) * ∫ z, f' z ∂Measure.pi (fun j => gaussianReal (m j) (v j)) := by
let μ : ι → Measure ℝ := fun j => gaussianReal (m j) (v j)
let ν := Measure.pi (fun j : {j // j ≠ i} => μ j)
let e := (Homeomorph.funSplitAt ℝ i).toMeasurableEquiv.symm
have hmp : MeasurePreserving e ((μ i).prod ν) (Measure.pi μ) :=
(measurePreserving_funSplitAt μ i).symm
have heval (x : ℝ × ({j // j ≠ i} → ℝ)) : e x i = x.1 := by
simp [e, Homeomorph.funSplitAt, Homeomorph.piSplitAt, Equiv.piSplitAt]
have hupdate (x : ℝ × ({j // j ≠ i} → ℝ)) (t : ℝ) :
Function.update (e x) i t = e (t, x.2) := by
ext j
by_cases hji : j = i
· subst j
simp [heval]
· simp [e, Homeomorph.funSplitAt,
Homeomorph.piSplitAt, Equiv.piSplitAt, hji]
have hxi : Integrable (fun z : ι → ℝ => z i - m i) (Measure.pi μ) := by
have hi : Integrable (fun r : ℝ => r - m i) (μ i) :=
((memLp_id_gaussianReal 1).integrable (by norm_num)).sub (integrable_const _)
exact (measurePreserving_eval μ i).integrable_comp_of_integrable hi
have hxfi : Integrable (fun z : ι → ℝ => (z i - m i) * f z) (Measure.pi μ) :=
hxi.mul_bdd hfc.aestronglyMeasurable (Filter.Eventually.of_forall fun z => by
simpa only [Real.norm_eq_abs] using hbound z)
have hf'i : Integrable f' (Measure.pi μ) :=
Integrable.of_bound hf'c.aestronglyMeasurable D
(Filter.Eventually.of_forall fun z => by
simpa only [Real.norm_eq_abs] using hbound' z)
have hprod := (hmp.integrable_comp_emb e.measurableEmbedding).2 hxfi
have hprod' := (hmp.integrable_comp_emb e.measurableEmbedding).2 hf'i
have hstein (y : {j // j ≠ i} → ℝ) :
(∫ r, (r - m i) * f (e (r, y)) ∂μ i) =
(v i : ℝ) * ∫ r, f' (e (r, y)) ∂μ i := by
apply gaussian_integral_centered_mul_of_bounded_deriv (m i) (v i)
· intro r
simpa only [hupdate] using hf (e (0, y)) r
· exact fun r => hbound _
· exact fun r => hbound' _
calc
_ = ∫ x, (e x i - m i) * f (e x) ∂(μ i).prod ν :=
(hmp.integral_comp e.measurableEmbedding _).symm
_ = ∫ y, ∫ r, (r - m i) * f (e (r, y)) ∂μ i ∂ν := by
simpa only [Function.comp_def, heval] using integral_prod_symm _ hprod
_ = ∫ y, (v i : ℝ) * ∫ r, f' (e (r, y)) ∂μ i ∂ν := by
congr 1
funext y
exact hstein y
_ = (v i : ℝ) * ∫ x, f' (e x) ∂(μ i).prod ν := by
rw [integral_const_mul]
congr 1
exact (integral_prod_symm _ hprod').symm
_ = _ := by rw [hmp.integral_comp e.measurableEmbedding]
theorem gaussian_pi_integral_mul_affine_of_bounded_deriv
(a b : ι → ℝ) (i : ι)
{f f' : (ι → ℝ) → ℝ} {C D : ℝ}
(hfc : Continuous f) (hf'c : Continuous f')
(hf : ∀ z t, HasDerivAt (fun r => f (Function.update z i r))
(f' (Function.update z i t)) t)
(hbound : ∀ z, |f z| ≤ C) (hbound' : ∀ z, |f' z| ≤ D) :
(∫ z, z i * f (fun j => a j + b j * z j)
∂Measure.pi (fun _ : ι => gaussianReal 0 1)) =
b i * ∫ z, f' (fun j => a j + b j * z j)
∂Measure.pi (fun _ : ι => gaussianReal 0 1) := by
let A : (ι → ℝ) → (ι → ℝ) := fun z j => a j + b j * z j
have hAc : Continuous A := continuous_pi fun j =>
continuous_const.add (continuous_const.mul (continuous_apply j))
have hupdate (z : ι → ℝ) (t : ℝ) :
A (Function.update z i t) = Function.update (A z) i (a i + b i * t) := by
ext j
by_cases hji : j = i
· subst j
simp [A]
· simp [A, hji]
have hd (z : ι → ℝ) (t : ℝ) :
HasDerivAt (fun r => f (A (Function.update z i r)))
(b i * f' (A (Function.update z i t))) t := by
simp only [hupdate]
simpa only [Function.comp_def, id_eq, mul_one, mul_comm] using
(hf (A z) (a i + b i * t)).comp t
(((hasDerivAt_id t).const_mul (b i)).const_add (a i))
have hb' (z : ι → ℝ) : |b i * f' (A z)| ≤ |b i| * D := by
rw [abs_mul]
exact mul_le_mul_of_nonneg_left (hbound' _) (abs_nonneg _)
have h := gaussian_pi_integral_centered_mul_of_bounded_deriv
(fun _ => 0) (fun _ => 1) i (hfc.comp hAc)
(continuous_const.mul (hf'c.comp hAc)) hd
(fun z => hbound (A z)) hb'
simpa only [Function.comp_def, Pi.mul_apply, A, sub_zero, NNReal.coe_one,
one_mul, integral_const_mul] using h
theorem gaussian_pi_integral_centered_mul_fderiv
(m : ι → ℝ) (v : ι → ℝ≥0) (i : ι)
{f : (ι → ℝ) → ℝ} {C D : ℝ}
(hf : ContDiff ℝ 1 f) (hbound : ∀ z, |f z| ≤ C)
(hbound' : ∀ z, |fderiv ℝ f z (Pi.single i 1)| ≤ D) :
(∫ z, (z i - m i) * f z ∂Measure.pi (fun j => gaussianReal (m j) (v j))) =
(v i : ℝ) * ∫ z, fderiv ℝ f z (Pi.single i 1)
∂Measure.pi (fun j => gaussianReal (m j) (v j)) := by
apply gaussian_pi_integral_centered_mul_of_bounded_deriv m v i hf.continuous
((hf.continuous_fderiv (by norm_num)).clm_apply continuous_const)
_ hbound hbound'
intro z t
exact ((hf.differentiable (by norm_num) _).hasFDerivAt).comp_hasDerivAt t
(hasDerivAt_update z i t)
theorem gaussian_pi_integral_mul_affine_fderiv
(a b : ι → ℝ) (i : ι)
{f : (ι → ℝ) → ℝ} {C D : ℝ}
(hf : ContDiff ℝ 1 f) (hbound : ∀ z, |f z| ≤ C)
(hbound' : ∀ z, |fderiv ℝ f z (Pi.single i 1)| ≤ D) :
(∫ z, z i * f (fun j => a j + b j * z j)
∂Measure.pi (fun _ : ι => gaussianReal 0 1)) =
b i * ∫ z, fderiv ℝ f (fun j => a j + b j * z j) (Pi.single i 1)
∂Measure.pi (fun _ : ι => gaussianReal 0 1) := by
apply gaussian_pi_integral_mul_affine_of_bounded_deriv a b i hf.continuous
((hf.continuous_fderiv (by norm_num)).clm_apply continuous_const)
_ hbound hbound'
intro z t
exact ((hf.differentiable (by norm_num) _).hasFDerivAt).comp_hasDerivAt t
(hasDerivAt_update z i t)
end CoordinateSplit
end Problem367
end
end OAI
|