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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 | import OAI.Probability.FreeIsing.Brownian.Observation
import OAI.Probability.FreeIsing.Posterior.PlantedBridgeInfinite
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367
def arrayNaturalFiltration {ι Ω : Type*} [mΩ : MeasurableSpace Ω]
(X : ι → ℝ≥0 → Ω → ℝ) (hm : ∀ i t, Measurable (X i t)) :
Filtration ℝ≥0 mΩ where
seq t := ⨆ i, Filtration.natural (X i) (fun t => (hm i t).stronglyMeasurable) t
mono' _ _ hst := iSup_mono fun i => (Filtration.natural (X i)
(fun t => (hm i t).stronglyMeasurable)).mono hst
le' t := iSup_le fun i => (Filtration.natural (X i)
(fun t => (hm i t).stronglyMeasurable)).le t
lemma arrayNaturalFiltration_eq_comap {ι Ω : Type*} [MeasurableSpace Ω]
(X : ι → ℝ≥0 → Ω → ℝ) (hm : ∀ i t, Measurable (X i t)) (t : ℝ≥0) :
arrayNaturalFiltration X hm t =
MeasurableSpace.comap (fun ω i (s : Set.Iic t) => X i s ω) inferInstance := by
simp only [arrayNaturalFiltration, MeasurableSpace.comap_process_pi,
Filtration.natural_eq_comap]
lemma arrayNaturalFiltration_stronglyAdapted {ι Ω : Type*} [MeasurableSpace Ω]
(X : ι → ℝ≥0 → Ω → ℝ) (hm : ∀ i t, Measurable (X i t)) (i : ι) :
StronglyAdapted (arrayNaturalFiltration X hm) (X i) := by
intro t
exact (Filtration.stronglyAdapted_natural
(fun t => (hm i t).stronglyMeasurable) t).mono (show
Filtration.natural (X i) (fun t => (hm i t).stronglyMeasurable) t ≤
⨆ j, Filtration.natural (X j) (fun t => (hm j t).stronglyMeasurable) t from
le_iSup (fun j : ι => Filtration.natural (X j)
(fun t => (hm j t).stronglyMeasurable) t) i)
lemma arrayNaturalFiltration_le {ι Ω : Type*} [mΩ : MeasurableSpace Ω]
(X : ι → ℝ≥0 → Ω → ℝ) (hm : ∀ i t, Measurable (X i t))
{F : Filtration ℝ≥0 mΩ} (had : ∀ i, StronglyAdapted F (X i)) :
arrayNaturalFiltration X hm ≤ F := by
intro t
exact iSup_le fun i => naturalFiltration_le_of_stronglyAdapted
(fun t => (hm i t).stronglyMeasurable) (had i) t
def signalArrayFiltration {ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω]
(B : ι → ℝ≥0 → Ω → ℝ) (hmB : ∀ i t, Measurable (B i t)) :
Filtration ℝ≥0 (inferInstance : MeasurableSpace (S × (ι → Ω))) :=
Filtration.const ℝ≥0 (MeasurableSpace.comap Prod.fst inferInstance)
measurable_fst.comap_le ⊔
arrayNaturalFiltration (fun i t (q : S × (ι → Ω)) => B i t (q.2 i))
(fun i t => (hmB i t).comp ((measurable_pi_apply i).comp measurable_snd))
lemma signalArrayFiltration_eq_comap {ι S Ω : Type*}
[MeasurableSpace S] [MeasurableSpace Ω]
(B : ι → ℝ≥0 → Ω → ℝ) (hmB : ∀ i t, Measurable (B i t)) (t : ℝ≥0) :
signalArrayFiltration (S := S) B hmB t =
MeasurableSpace.comap
(fun q : S × (ι → Ω) => (q.1, fun i (s : Set.Iic t) => B i s (q.2 i)))
inferInstance := by
change MeasurableSpace.comap Prod.fst inferInstance ⊔
arrayNaturalFiltration _ _ t = _
rw [arrayNaturalFiltration_eq_comap]
exact (MeasurableSpace.comap_prodMk Prod.fst
(fun q : S × (ι → Ω) => fun i (s : Set.Iic t) => B i s (q.2 i))).symm
theorem planted_array_increments_indep_signal_past
{ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω]
(ν : Measure S) (μ : ι → Measure Ω) [IsProbabilityMeasure ν]
{B : ι → ℝ≥0 → Ω → ℝ} (hB : ∀ i, IsPreBrownianReal (B i) (μ i))
(hmB : ∀ i t, Measurable (B i t)) {s t : ℝ≥0} (hst : s ≤ t) :
IndepFun
(fun q : S × (ι → Ω) => fun i => B i t (q.2 i) - B i s (q.2 i))
(fun q : S × (ι → Ω) => (q.1, fun i (u : Set.Iic s) => B i u (q.2 i)))
(ν.prod (Measure.infinitePi μ)) := by
have (i : ι) : IsProbabilityMeasure (μ i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
have hmPast (i : ι) : Measurable (fun ω (u : Set.Iic s) => B i u ω) :=
Measurable.of_eval fun u => hmB i u
have hmInc (i : ι) : Measurable (fun ω => B i t ω - B i s ω) :=
(hmB i t).sub (hmB i s)
have hind := indepFun_infinitePi_of_coordinate_indepFun μ hmPast hmInc
(fun i => (preBrownian_increment_indep_past (hB i) hst).symm)
exact (indepFun_prod_left_of_indepFun ν (Measure.infinitePi μ)
(Measurable.of_eval fun i => (hmPast i).comp (measurable_pi_apply i))
(Measurable.of_eval fun i => (hmInc i).comp (measurable_pi_apply i)) hind).symm
lemma planted_noise_coordinate_measurePreserving
{ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω]
(ν : Measure S) (μ : ι → Measure Ω) [IsProbabilityMeasure ν]
[∀ i, IsProbabilityMeasure (μ i)] (i : ι) :
MeasurePreserving (fun q : S × (ι → Ω) => q.2 i)
(ν.prod (Measure.infinitePi μ)) (μ i) := by
refine ⟨(measurable_pi_apply i).comp measurable_snd, ?_⟩
change Measure.map ((fun w : ι → Ω => w i) ∘ Prod.snd) _ = _
rw [← Measure.map_map (measurable_pi_apply i) measurable_snd]
simp only [Measure.map_snd_prod, measure_univ, one_smul]
exact Measure.infinitePi_map_eval μ i
theorem preBrownian_martingale_signalArrayFiltration
{ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω]
(ν : Measure S) (μ : ι → Measure Ω) [IsProbabilityMeasure ν]
{B : ι → ℝ≥0 → Ω → ℝ} (hB : ∀ i, IsPreBrownianReal (B i) (μ i))
(hmB : ∀ i t, Measurable (B i t)) (i : ι) :
Martingale (fun t (q : S × (ι → Ω)) => B i t (q.2 i))
(signalArrayFiltration B hmB) (ν.prod (Measure.infinitePi μ)) := by
have (j : ι) : IsProbabilityMeasure (μ j) :=
(hB j).isGaussianProcess.isProbabilityMeasure
let P := ν.prod (Measure.infinitePi μ)
let F := signalArrayFiltration (S := S) B hmB
have hproj := planted_noise_coordinate_measurePreserving ν μ i
have hint (t : ℝ≥0) : Integrable (fun q : S × (ι → Ω) => B i t (q.2 i)) P :=
hproj.integrable_comp_of_integrable ((hB i).integrable_eval t)
have had : StronglyAdapted F (fun t (q : S × (ι → Ω)) => B i t (q.2 i)) := by
intro t
exact (arrayNaturalFiltration_stronglyAdapted
(fun j u (q : S × (ι → Ω)) => B j u (q.2 j))
(fun j u => (hmB j u).comp ((measurable_pi_apply j).comp measurable_snd)) i t).mono
le_sup_right
apply martingale_of_conditional_increment_zero had hint
intro s t hst
have hind := (planted_array_increments_indep_signal_past ν μ hB hmB hst).comp
(measurable_pi_apply i) measurable_id
have hn := signalArrayFiltration_eq_comap (S := S) B hmB s
have hcond : P[(fun q : S × (ι → Ω) => B i t (q.2 i) - B i s (q.2 i)) | F s] =ᵐ[P]
fun _ => ∫ q : S × (ι → Ω), B i t (q.2 i) - B i s (q.2 i) ∂P := by
change P[_ | signalArrayFiltration B hmB s] =ᵐ[P] _
rw [hn]
exact condExp_indep_eq
(((hmB i t).sub (hmB i s)).comp hproj.measurable).comap_le
(hn ▸ F.le s) (measurable_iff_comap_le.mpr le_rfl).stronglyMeasurable hind
have hzero : (∫ q : S × (ι → Ω), B i t (q.2 i) - B i s (q.2 i) ∂P) = 0 := by
have hi := integral_map (μ := P) hproj.measurable.aemeasurable
(((hmB i t).sub (hmB i s)).aestronglyMeasurable)
rw [hproj.map_eq] at hi
simp only [Pi.sub_apply] at hi
rw [← hi, integral_sub ((hB i).integrable_eval t) ((hB i).integrable_eval s),
(hB i).integral_eval, (hB i).integral_eval, sub_self]
simpa only [hzero, Pi.sub_def] using hcond
def plantedArrayObservation {ι S Ω : Type*}
(B : ι → ℝ≥0 → Ω → ℝ) (σ : S → ι → ℝ)
(i : ι) (t : ℝ≥0) (q : S × (ι → Ω)) : ℝ :=
B i t (q.2 i) + (t : ℝ) * σ q.1 i
lemma plantedArrayObservation_measurable {ι S Ω : Type*}
[MeasurableSpace S] [MeasurableSpace Ω]
{B : ι → ℝ≥0 → Ω → ℝ} (hmB : ∀ i t, Measurable (B i t))
{σ : S → ι → ℝ} (hσ : Measurable σ) (i : ι) (t : ℝ≥0) :
Measurable (plantedArrayObservation B σ i t) :=
((hmB i t).comp ((measurable_pi_apply i).comp measurable_snd)).add
((((measurable_pi_apply i).comp hσ).comp measurable_fst).const_mul _)
def arrayObservationFiltration {ι S Ω : Type*}
[MeasurableSpace S] [MeasurableSpace Ω]
(B : ι → ℝ≥0 → Ω → ℝ) (hmB : ∀ i t, Measurable (B i t))
(σ : S → ι → ℝ) (hσ : Measurable σ) :
Filtration ℝ≥0 (inferInstance : MeasurableSpace (S × (ι → Ω))) :=
arrayNaturalFiltration (plantedArrayObservation B σ)
(plantedArrayObservation_measurable hmB hσ)
lemma arrayObservationFiltration_le_signalArrayFiltration {ι S Ω : Type*}
[MeasurableSpace S] [MeasurableSpace Ω]
(B : ι → ℝ≥0 → Ω → ℝ) (hmB : ∀ i t, Measurable (B i t))
(σ : S → ι → ℝ) (hσ : Measurable σ) :
arrayObservationFiltration B hmB σ hσ ≤ signalArrayFiltration B hmB := by
apply arrayNaturalFiltration_le
intro i t
have hnoise : StronglyMeasurable[signalArrayFiltration (S := S) B hmB t]
(fun q : S × (ι → Ω) => B i t (q.2 i)) :=
(arrayNaturalFiltration_stronglyAdapted
(fun j u (q : S × (ι → Ω)) => B j u (q.2 j))
(fun j u => (hmB j u).comp ((measurable_pi_apply j).comp measurable_snd)) i t).mono
le_sup_right
have hsignal : StronglyMeasurable[MeasurableSpace.comap
(Prod.fst : S × (ι → Ω) → S) inferInstance]
(fun q : S × (ι → Ω) => σ q.1 i) :=
(((measurable_pi_apply i).comp hσ).comp (comap_measurable Prod.fst)).stronglyMeasurable
exact hnoise.add (stronglyMeasurable_const.mul (hsignal.mono le_sup_left))
theorem preBrownian_array_observation_innovation_martingale
{ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω]
(ν : Measure S) (μ : ι → Measure Ω) [IsProbabilityMeasure ν]
{B : ι → ℝ≥0 → Ω → ℝ} (hB : ∀ i, IsPreBrownianReal (B i) (μ i))
(hmB : ∀ i t, Measurable (B i t))
{σ : S → ι → ℝ} (hσ : Measurable σ) (i : ι)
(hσint : Integrable (fun s => σ s i) ν)
{m : ℝ≥0 → S × (ι → Ω) → ℝ}
(hmAd : StronglyAdapted (arrayObservationFiltration B hmB σ hσ) m)
(hposterior : ∀ t, m t =ᵐ[ν.prod (Measure.infinitePi μ)]
(ν.prod (Measure.infinitePi μ))[(fun q : S × (ι → Ω) => σ q.1 i) |
arrayObservationFiltration B hmB σ hσ t])
(hAad : StronglyAdapted (arrayObservationFiltration B hmB σ hσ)
(nnrealTimePrimitive m))
(hInt : ∀ s t : ℝ≥0, s ≤ t →
Integrable (fun p : ℝ × (S × (ι → Ω)) => m p.1.toNNReal p.2)
((volume.restrict (Set.Ioc (s : ℝ) (t : ℝ))).prod
(ν.prod (Measure.infinitePi μ)))) :
Martingale
(fun t q => plantedArrayObservation B σ i t q - nnrealTimePrimitive m t q)
(arrayObservationFiltration B hmB σ hσ) (ν.prod (Measure.infinitePi μ)) := by
have (j : ι) : IsProbabilityMeasure (μ j) :=
(hB j).isGaussianProcess.isProbabilityMeasure
exact compensated_observation_martingale_of_posterior_version
(preBrownian_martingale_signalArrayFiltration ν μ hB hmB i)
(arrayObservationFiltration_le_signalArrayFiltration B hmB σ hσ)
(hσint.comp_fst (Measure.infinitePi μ))
(arrayNaturalFiltration_stronglyAdapted (plantedArrayObservation B σ)
(plantedArrayObservation_measurable hmB hσ) i)
hmAd hposterior hAad hInt
end Problem367
end
end OAI
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