File size: 3,538 Bytes
b317718 | 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 | import OAI.Probability.FreeIsing.Posterior.FilteringCrossLimit
import OAI.Probability.FreeIsing.Posterior.ProgressiveIsingPosterior
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367
theorem stronglyMeasurable_latent_signalArrayFiltration
{ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω]
(B : ι → ℝ≥0 → Ω → ℝ) (hmB : ∀ i t, Measurable (B i t))
{Q : S → ℝ} (hQ : Measurable Q) (t : ℝ≥0) :
StronglyMeasurable[signalArrayFiltration B hmB t]
(fun q : S × (ι → Ω) => Q q.1) := by
have hm : StronglyMeasurable[MeasurableSpace.comap
(Prod.fst : S × (ι → Ω) → S) inferInstance] (fun q => Q q.1) :=
(hQ.comp (comap_measurable Prod.fst)).stronglyMeasurable
exact hm.mono le_sup_left
theorem planted_array_posterior_uniformSum_cross_energy
{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)
{Q : S → ℝ} (hQ : Measurable Q) {A C D : ℝ}
(hQb : ∀ᵐ q ∂ν, |Q q| ≤ D)
(hσint : Integrable (fun q => σ q i) ν)
(hσb : ∀ᵐ q ∂ν, |σ q i| ≤ C)
{m K : ℝ≥0 → S × (ι → Ω) → ℝ}
(hm : IsStronglyProgressive (arrayObservationFiltration B hmB σ hσ) m)
(hpost : ∀ r, m r =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 r])
(hK : StronglyAdapted (arrayObservationFiltration B hmB σ hσ) K)
(hKc : ∀ᵐ q ∂ν.prod (Measure.infinitePi μ), Continuous (fun r => K r q))
(hA : 0 ≤ A)
(hKb : ∀ᵐ q ∂ν.prod (Measure.infinitePi μ), ∀ r, |K r q| ≤ A)
(hKversion : ∀ r, K r =ᵐ[ν.prod (Measure.infinitePi μ)] fun q =>
(ν.prod (Measure.infinitePi μ))[(fun q : S × (ι → Ω) => Q q.1 * σ q.1 i) |
arrayObservationFiltration B hmB σ hσ r] q -
(ν.prod (Measure.infinitePi μ))[(fun q : S × (ι → Ω) => Q q.1) |
arrayObservationFiltration B hmB σ hσ r] q * m r q)
(T : ℝ≥0) :
Tendsto (fun n : ℕ => ∫ q,
(ν.prod (Measure.infinitePi μ))[(fun q : S × (ι → Ω) => Q q.1) |
arrayObservationFiltration B hmB σ hσ T] q *
BrownianGrid.uniformSum
(fun t q => plantedArrayObservation B σ i t q - nnrealTimePrimitive m t q)
K T n q ∂ν.prod (Measure.infinitePi μ)) atTop
(𝓝 (∫ r in (0 : ℝ)..(T : ℝ),
∫ q, K r.toNNReal q ^ 2 ∂ν.prod (Measure.infinitePi μ))) := by
have (index : ι) : IsProbabilityMeasure (μ index) :=
(hB index).isGaussianProcess.isProbabilityMeasure
exact FilteringCrossEnergy.tendsto_posterior_uniformSum_cross_energy
(preBrownian_martingale_signalArrayFiltration ν μ hB hmB i)
(arrayObservationFiltration_le_signalArrayFiltration B hmB σ hσ)
(stronglyMeasurable_latent_signalArrayFiltration B hmB hQ 0)
(Measure.quasiMeasurePreserving_fst.ae hQb)
(hσint.comp_fst (Measure.infinitePi μ))
(Measure.quasiMeasurePreserving_fst.ae hσb)
(arrayNaturalFiltration_stronglyAdapted (plantedArrayObservation B σ)
(plantedArrayObservation_measurable hmB hσ) i)
hm hpost hK hKc hA hKb hKversion T
end Problem367
end
end OAI
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