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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 | import OAI.Probability.FreeIsing.Brownian.ArrayObservation
import OAI.Probability.FreeIsing.Stochastic.SimpleIntegralIsometry
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367
theorem planted_coordinate_preBrownian
{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)) (i : ι) :
IsPreBrownianReal (fun t (q : S × (ι → Ω)) => B i t (q.2 i))
(ν.prod (Measure.infinitePi μ)) where
hasLaw I := by
have (j : ι) : IsProbabilityMeasure (μ j) :=
(hB j).isGaussianProcess.isProbabilityMeasure
exact ((hB i).hasLaw I).comp
(planted_noise_coordinate_measurePreserving ν μ i).hasLaw
theorem planted_coordinate_increments_indep
{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) {i j : ι} (hij : i ≠ j) :
IndepFun (fun q : S × (ι → Ω) => B i t (q.2 i) - B i s (q.2 i))
(fun q : S × (ι → Ω) => B j t (q.2 j) - B j s (q.2 j))
(ν.prod (Measure.infinitePi μ)) := by
have (k : ι) : IsProbabilityMeasure (μ k) :=
(hB k).isGaussianProcess.isProbabilityMeasure
have hind := (iIndepFun_infinitePi
(fun k => (hmB k t).sub (hmB k s)) (P := μ)).indepFun hij
exact indepFun_comp_measurePreserving measurePreserving_snd
(((hmB i t).sub (hmB i s)).comp (measurable_pi_apply i))
(((hmB j t).sub (hmB j s)).comp (measurable_pi_apply j)) hind
theorem planted_weighted_increment_integral_sq
{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)
(I : Finset ι) (a : ι → ℝ) :
(∫ q : S × (ι → Ω),
(∑ i ∈ I, a i * (B i t (q.2 i) - B i s (q.2 i))) ^ 2
∂ν.prod (Measure.infinitePi μ)) =
((t : ℝ) - (s : ℝ)) * ∑ i ∈ I, a i ^ 2 := by
classical
have (k : ι) : IsProbabilityMeasure (μ k) :=
(hB k).isGaussianProcess.isProbabilityMeasure
let P := ν.prod (Measure.infinitePi μ)
let D : ι → S × (ι → Ω) → ℝ := fun i q => B i t (q.2 i) - B i s (q.2 i)
have hC (i : ι) := planted_coordinate_preBrownian ν μ hB i
have hD2 (i : ι) : MemLp (D i) 2 P :=
((hC i).isGaussianProcess.hasGaussianLaw_eval t).memLp_two.sub
((hC i).isGaussianProcess.hasGaussianLaw_eval s).memLp_two
have hY2 (i : ι) : MemLp (fun q => a i * D i q) 2 P := (hD2 i).const_mul (a i)
have hzero (i : ι) : (∫ q, D i q ∂P) = 0 := by
change (∫ q : S × (ι → Ω), B i t (q.2 i) - B i s (q.2 i) ∂P) = 0
rw [integral_sub ((hC i).integrable_eval t) ((hC i).integrable_eval s), (hC i).integral_eval, (hC i).integral_eval, sub_self]
have hsq (i : ι) : (∫ q, D i q ^ 2 ∂P) = (t : ℝ) - (s : ℝ) := by
have h := preBrownian_integral_sq ((hC i).shift s) (t - s)
simpa only [D, add_tsub_cancel_of_le hst, NNReal.coe_sub hst] using h
have horth (i : ι) (_hi : i ∈ I) (j : ι) (_hj : j ∈ I) (hij : i ≠ j) :
(∫ q, (a i * D i q) * (a j * D j q) ∂P) = 0 := by
have hind := (planted_coordinate_increments_indep ν μ hB hmB s t hij).comp
(measurable_id.const_mul (a i)) (measurable_id.const_mul (a j))
have heq := hind.integral_mul_eq_mul_integral
(hY2 i).aestronglyMeasurable (hY2 j).aestronglyMeasurable
change (∫ q, (a i * D i q) * (a j * D j q) ∂P) =
(∫ q, a i * D i q ∂P) * (∫ q, a j * D j q ∂P) at heq
simpa only [integral_const_mul, hzero, mul_zero, zero_mul] using heq
rw [SimpleIntegral.integral_sq_sum_of_orthogonal I (fun i q => a i * D i q) (fun i _ j _ => (hY2 i).integrable_mul (hY2 j)) horth]
simp only [mul_pow, integral_const_mul, hsq]
rw [Finset.mul_sum]
exact Finset.sum_congr rfl fun i _ => mul_comm _ _
theorem planted_weighted_increment_sq_condExp
{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)
(I : Finset ι) (a : ι → ℝ) :
(ν.prod (Measure.infinitePi μ))[ (fun q : S × (ι → Ω) => (∑ i ∈ I, a i * (B i t (q.2 i) - B i s (q.2 i))) ^ 2) | signalArrayFiltration B hmB s] =ᵐ[ν.prod (Measure.infinitePi μ)]
fun _ => ((t : ℝ) - (s : ℝ)) * ∑ i ∈ I, a i ^ 2 := by
classical
have (k : ι) : IsProbabilityMeasure (μ k) :=
(hB k).isGaussianProcess.isProbabilityMeasure
have hm : Measurable (fun x : ι → ℝ => (∑ i ∈ I, a i * x i) ^ 2) := by
fun_prop
have hind := (planted_array_increments_indep_signal_past ν μ hB hmB hst).comp
hm measurable_id
have hf : Measurable (fun q : S × (ι → Ω) =>
(∑ i ∈ I, a i * (B i t (q.2 i) - B i s (q.2 i))) ^ 2) := by
apply hm.comp
exact Measurable.of_eval fun i =>
((hmB i t).sub (hmB i s)).comp ((measurable_pi_apply i).comp measurable_snd)
have hn := signalArrayFiltration_eq_comap (S := S) B hmB s
have hc := condExp_indep_eq hf.comap_le
(hn ▸ (signalArrayFiltration (S := S) B hmB).le s)
(measurable_iff_comap_le.mpr le_rfl).stronglyMeasurable hind
simp only [Function.id_comp] at hc
rw [← hn] at hc
simpa only [Function.comp_def, id_eq, planted_weighted_increment_integral_sq ν μ hB hmB hst I a] using hc
theorem planted_weighted_increment_sq_condExp_subfield
{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)
(I : Finset ι) (a : ι → ℝ)
{m : MeasurableSpace (S × (ι → Ω))} (hm : m ≤ signalArrayFiltration B hmB s) :
(ν.prod (Measure.infinitePi μ))[ (fun q : S × (ι → Ω) => (∑ i ∈ I, a i * (B i t (q.2 i) - B i s (q.2 i))) ^ 2) | m]
=ᵐ[ν.prod (Measure.infinitePi μ)]
fun _ => ((t : ℝ) - (s : ℝ)) * ∑ i ∈ I, a i ^ 2 := by
have (k : ι) : IsProbabilityMeasure (μ k) :=
(hB k).isGaussianProcess.isProbabilityMeasure
exact ((condExp_condExp_of_le hm (signalArrayFiltration (S := S) B hmB |>.le s)).symm.trans
(condExp_congr_ae (planted_weighted_increment_sq_condExp ν μ hB hmB hst I a))).trans
(Filter.EventuallyEq.of_eq (condExp_const (hm.trans
(signalArrayFiltration (S := S) B hmB |>.le s)) _))
end Problem367
end
end OAI
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