File size: 7,154 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 | import OAI.Probability.FreeIsing.Brownian.Observation
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.InnovationConditionalMoments
variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
theorem integral_abs_condExp_sub_const_le [IsFiniteMeasure P]
{m : MeasurableSpace Ω} {X Y : Ω → ℝ} {c : ℝ}
(hX : Integrable X P) (hY : Integrable Y P)
(hmean : P[Y | m] =ᵐ[P] fun _ => c) :
(∫ ω, |P[X | m] ω - c| ∂P) ≤ ∫ ω, |X ω - Y ω| ∂P := by
have heq : (fun ω => P[X | m] ω - c) =ᵐ[P] P[X - Y | m] := by
filter_upwards [condExp_sub hX hY m, hmean] with ω hsub hy
simpa only [Pi.sub_apply, hy] using hsub.symm
calc
(∫ ω, |P[X | m] ω - c| ∂P) = ∫ ω, |P[X - Y | m] ω| ∂P :=
integral_congr_ae (heq.fun_comp abs)
_ ≤ ∫ ω, |X ω - Y ω| ∂P := by
simpa only [Real.norm_eq_abs, Pi.sub_apply] using
(integral_norm_condExp_le (μ := P) (m := m) (X - Y))
theorem integral_smul_eq_integral_condExp_smul [IsFiniteMeasure P]
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
{m : MeasurableSpace Ω} (hm : m ≤ mΩ) {f : Ω → ℝ} (hf : Integrable f P)
{Z : Ω → E} (hZ : StronglyMeasurable[m] Z) {C : ℝ}
(hbound : ∀ᵐ ω ∂P, ‖Z ω‖ ≤ C) :
(∫ ω, f ω • Z ω ∂P) = ∫ ω, P[f | m] ω • Z ω ∂P := by
have hprod : Integrable (f • Z) P :=
hf.smul_bdd C (hZ.mono hm).aestronglyMeasurable hbound
calc
(∫ ω, f ω • Z ω ∂P) = ∫ ω, P[f • Z | m] ω ∂P :=
(integral_condExp hm).symm
_ = ∫ ω, P[f | m] ω • Z ω ∂P :=
integral_congr_ae
(condExp_smul_of_aestronglyMeasurable_right hf hprod hZ.aestronglyMeasurable)
theorem norm_integral_smul_le_conditional_abs [IsFiniteMeasure P]
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
{m : MeasurableSpace Ω} (hm : m ≤ mΩ) {f : Ω → ℝ} (hf : Integrable f P)
{Z : Ω → E} (hZ : StronglyMeasurable[m] Z) {C : ℝ}
(hbound : ∀ᵐ ω ∂P, ‖Z ω‖ ≤ C) :
‖∫ ω, f ω • Z ω ∂P‖ ≤ C * ∫ ω, |P[f | m] ω| ∂P := by
rw [integral_smul_eq_integral_condExp_smul (mΩ := mΩ) hm hf hZ hbound]
calc
‖∫ ω, P[f | m] ω • Z ω ∂P‖ ≤ ∫ ω, C * |P[f | m] ω| ∂P := by
apply norm_integral_le_of_norm_le (integrable_condExp.abs.const_mul C)
filter_upwards [hbound] with ω hω
rw [norm_smul, Real.norm_eq_abs]
nlinarith [abs_nonneg (P[f | m] ω)]
_ = C * ∫ ω, |P[f | m] ω| ∂P := integral_const_mul _ _
theorem norm_integral_test_sub_const_le [IsFiniteMeasure P]
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
{m : MeasurableSpace Ω} (hm : m ≤ mΩ) {X Y : Ω → ℝ} {c : ℝ}
(hX : Integrable X P) (hY : Integrable Y P)
(hmean : P[Y | m] =ᵐ[P] fun _ => c)
{Z : Ω → E} (hZ : StronglyMeasurable[m] Z) {C : ℝ} (hC : 0 ≤ C)
(hbound : ∀ᵐ ω ∂P, ‖Z ω‖ ≤ C) :
‖∫ ω, (X ω - c) • Z ω ∂P‖ ≤ C * ∫ ω, |X ω - Y ω| ∂P := by
have hce : P[(fun ω => X ω - c) | m] =ᵐ[P]
fun ω => P[X | m] ω - c := by
have h := condExp_sub hX (integrable_const c) m
rw [condExp_const hm] at h
exact h
calc
‖∫ ω, (X ω - c) • Z ω ∂P‖ ≤
C * ∫ ω, |P[(fun ω => X ω - c) | m] ω| ∂P :=
norm_integral_smul_le_conditional_abs (mΩ := mΩ) hm
(hX.sub (integrable_const c)) hZ hbound
_ = C * ∫ ω, |P[X | m] ω - c| ∂P := by
congr 1
exact integral_congr_ae (hce.fun_comp abs)
_ ≤ C * ∫ ω, |X ω - Y ω| ∂P :=
mul_le_mul_of_nonneg_left
(integral_abs_condExp_sub_const_le (mΩ := mΩ) hX hY hmean) hC
theorem preBrownian_increment_sq_condExp_enlarged
{B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P)
(hmB : ∀ t, StronglyMeasurable (B t))
{mSignal m : MeasurableSpace Ω} (hSignal : mSignal ≤ mΩ)
(hind : Indep (MeasurableSpace.comap (fun ω t => B t ω) inferInstance) mSignal P)
{s t : ℝ≥0} (hst : s ≤ t)
(hm : m ≤ Filtration.natural B hmB s ⊔ mSignal) :
P[(fun ω => (B t ω - B s ω)^2) | m] =ᵐ[P]
fun _ => (t : ℝ) - (s : ℝ) := by
let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
let mPath := MeasurableSpace.comap (fun ω t => B t ω) inferInstance
let : MeasurableSpace Ω := mΩ
have hpath : mPath ≤ mΩ :=
(Measurable.of_eval (fun time => (hmB time).measurable)).comap_le
have hcoord (u : ℝ≥0) : StronglyMeasurable[mPath] (B u) := by
have harray : @Measurable Ω (ℝ≥0 → ℝ) mPath inferInstance
(fun ω t => B t ω) := measurable_iff_comap_le.mpr le_rfl
exact ((measurable_pi_apply u).comp harray).stronglyMeasurable
have hnat : Filtration.natural B hmB s ≤ mPath := by
change (⨆ u ≤ s, MeasurableSpace.comap (B u) inferInstance) ≤ mPath
exact iSup₂_le fun u _ => (hcoord u).measurable.comap_le
have hint : Integrable (fun ω => (B t ω - B s ω)^2) P :=
((hB.isGaussianProcess.hasGaussianLaw_eval t).memLp_two.sub
(hB.isGaussianProcess.hasGaussianLaw_eval s).memLp_two).integrable_sq
have hlarge : P[(fun ω => (B t ω - B s ω)^2) |
Filtration.natural B hmB s ⊔ mSignal] =ᵐ[P]
fun _ => (t : ℝ) - (s : ℝ) :=
(condExp_sup_independent_future hnat hpath hSignal hind
(((hcoord t).sub (hcoord s)).pow 2) hint).trans
(preBrownian_increment_sq_condExp (mΩ := mΩ) hB hmB hst)
have hlarge_le : Filtration.natural B hmB s ⊔ mSignal ≤ mΩ :=
sup_le (Filtration.natural B hmB |>.le s) hSignal
calc
P[(fun ω => (B t ω - B s ω)^2) | m] =ᵐ[P]
P[P[(fun ω => (B t ω - B s ω)^2) |
Filtration.natural B hmB s ⊔ mSignal] | m] :=
(condExp_condExp_of_le hm hlarge_le).symm
_ =ᵐ[P] P[(fun _ : Ω => (t : ℝ) - (s : ℝ)) | m] :=
condExp_congr_ae hlarge
_ =ᵐ[P] fun _ => (t : ℝ) - (s : ℝ) := by
rw [condExp_const (hm.trans hlarge_le)]
theorem preBrownian_integral_abs_conditional_sq_error_le
{B : ℝ≥0 → Ω → ℝ} (hB : IsPreBrownianReal B P)
(hmB : ∀ t, StronglyMeasurable (B t))
{mSignal m : MeasurableSpace Ω} (hSignal : mSignal ≤ mΩ)
(hind : Indep (MeasurableSpace.comap (fun ω t => B t ω) inferInstance) mSignal P)
{s t : ℝ≥0} (hst : s ≤ t)
(hm : m ≤ Filtration.natural B hmB s ⊔ mSignal)
{D : Ω → ℝ} (hD : MemLp D 2 P) :
(∫ ω, |P[(fun ω => D ω ^ 2) | m] ω - ((t : ℝ) - (s : ℝ))| ∂P) ≤
∫ ω, |D ω ^ 2 - (B t ω - B s ω)^2| ∂P := by
let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
exact integral_abs_condExp_sub_const_le (mΩ := mΩ) hD.integrable_sq
((hB.isGaussianProcess.hasGaussianLaw_eval t).memLp_two.sub
(hB.isGaussianProcess.hasGaussianLaw_eval s).memLp_two).integrable_sq
(preBrownian_increment_sq_condExp_enlarged (mΩ := mΩ) hB hmB hSignal hind hst hm)
end Problem367.InnovationConditionalMoments
end
end OAI
|