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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 | import OAI.Probability.FreeIsing.Conditioning.ContinuousCopyImmersion
import OAI.Probability.FreeIsing.Ising.CanonicalMartingales
import OAI.Probability.FreeIsing.Ising.CopyBrownian
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation ContinuousObservationHistory ContinuousCopyBrownian
variable {d : ℕ} {beta : ℝ}
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
lemma stoppedArray_comap_eq_IsingFiltration (t : ℝ≥0) :
MeasurableSpace.comap (stoppedArray (ι := TreeVertex d) t) inferInstance = filtration d t := by
exact stoppedArray_comap_eq_natural t
lemma measurable_observation_singleFiltration (t : ℝ≥0) :
Measurable[singleFiltration (ι := TreeVertex d) t, filtration d t]
(Prod.snd : Paths d × Paths d → Paths d) := by
apply Measurable.of_comap_le
rw [← stoppedArray_comap_eq_IsingFiltration, MeasurableSpace.comap_comp]
exact (measurable_snd.comp (comap_measurable (singleHistory t))).comap_le
lemma graphPast_comap_le (beta : ℝ) (t : ℝ≥0) :
(singleFiltration (ι := TreeVertex d) t).comap
(fun x : Paths d => (innovationMap beta x, x)) ≤ filtration d t := by
change (MeasurableSpace.comap (singleHistory t)
(inferInstance : MeasurableSpace (Paths d × Paths d))).comap
(fun x : Paths d => (innovationMap beta x, x)) ≤ _
rw [MeasurableSpace.comap_comp]
have hW : Measurable[filtration d t]
(fun x : Paths d => stoppedArray t (innovationMap beta x)) :=
measurable_stoppedArray_of_eval (mΩ := filtration d t) t (innovationMap beta)
(fun v s hs => ((innovationMap_stronglyAdapted beta v s).measurable).mono
((filtration d).mono hs) le_rfl)
have hX : Measurable[filtration d t] (stoppedArray (ι := TreeVertex d) t) := by
rw [← stoppedArray_comap_eq_IsingFiltration]
exact comap_measurable _
exact (hW.prodMk hX).comap_le
theorem martingale_graph_of_observation
{M : ℝ≥0 → Paths d → ℝ}
(hM : Martingale M (filtration d) (observationLaw μ)) :
Martingale (fun t (q : Paths d × Paths d) => M t q.2)
singleFiltration (((observationLaw μ).map (innovationMap beta)) ⊗ₘ kernel beta μ) := by
rw [kernel_graphLaw]
have hgraph : MeasurePreserving (fun x : Paths d => (innovationMap beta x, x))
(observationLaw μ) (IsingConditionalCopyBounds.graphLaw beta μ) :=
⟨(measurable_innovationMap beta).prodMk measurable_id, rfl⟩
exact MartingalePushforward.martingale_of_pullback hgraph
(F := filtration d) (G := singleFiltration) (graphPast_comap_le beta)
(fun t => (hM.stronglyAdapted t).comp_measurable
(measurable_observation_singleFiltration t)) hM
theorem martingale_copy_projection
(hμ : HasFreeIsingLaw (Real.tanh beta) μ)
{M : ℝ≥0 → Paths d → ℝ}
(hM : Martingale M (filtration d) (observationLaw μ)) (j : Bool) :
Martingale (fun t (q : IsingCopyEnergy.Joint d) => M t (IsingCopyEnergy.projection j q))
ContinuousCopyBrownian.joinedFiltration (copies beta μ) := by
have : IsProbabilityMeasure ((observationLaw μ).map (innovationMap beta)) :=
inferInstance
have hg := martingale_graph_of_observation (beta := beta) hM
cases j
· exact martingale_right_of_future_indep _ _ (IsingCopyBrownian.graph_future_indep hμ) hg
· exact martingale_left_of_future_indep _ _ (IsingCopyBrownian.graph_future_indep hμ) hg
theorem copyContinuousPosterior_martingale
(hd : 3 ≤ d) (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(v : TreeVertex d) (j : Bool) :
Martingale (copyContinuousPosterior beta v j)
ContinuousCopyBrownian.joinedFiltration (copies beta μ) :=
martingale_copy_projection hμ (continuousPosterior_martingale hd hμ hb hcrit v) j
theorem copyContinuousPosterior_compensatedSquare_martingale
(hd : 3 ≤ d) (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(v : TreeVertex d) (j : Bool) :
Martingale (fun t (q : IsingCopyEnergy.Joint d) => copyContinuousPosterior beta v j t q ^ 2 -
canonicalCovarianceCompensator beta v t (IsingCopyEnergy.projection j q))
ContinuousCopyBrownian.joinedFiltration (copies beta μ) :=
martingale_copy_projection hμ
(continuousPosterior_compensatedSquare_martingale hd hμ hb hcrit v) j
end Problem367.IsingObservationFactor
end
end OAI
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