File size: 5,977 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 | import OAI.Probability.FreeIsing.Estimates.ProgressiveVersionPushforward
import OAI.Probability.FreeIsing.Ising.ContinuousPosterior
import OAI.Probability.FreeIsing.Ising.CopyStoppedEnergy
import OAI.Probability.FreeIsing.Posterior.ContinuousPosteriorInterior
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation
def continuousField {d : ℕ} (beta : ℝ) (v : TreeVertex d) : ℝ≥0 → Paths d → ℝ :=
ProgressiveVersionPushforward.version (field beta v)
lemma field_progressive {d : ℕ} (beta : ℝ) (v : TreeVertex d) :
IsStronglyProgressive (filtration d) (field beta v) :=
isStronglyProgressive_artanh (filtration d) (posterior_progressive beta v)
lemma continuousField_progressive {d : ℕ} (beta : ℝ) (v : TreeVertex d) :
IsStronglyProgressive (filtration d) (continuousField beta v) :=
ProgressiveVersionPushforward.progressive_version (field_progressive beta v)
theorem continuousField_modification
{d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (v : TreeVertex d) :
(∀ᵐ x ∂observationLaw μ, Continuous (fun t => continuousField beta v t x)) ∧
(∀ t, continuousField beta v t =ᵐ[observationLaw μ] field beta v t) := by
let B : ℝ≥0 → ℝ → ℝ := fun t u => WienerSampler.sample u t
have hmB (t : ℝ≥0) : Measurable (B t) :=
(continuous_eval_const t).measurable.comp WienerSampler.measurable_sample
have hcB (u : ℝ) : Continuous (fun t => B t u) := (WienerSampler.sample u).continuous
obtain ⟨Z, hZ⟩ := Classical.axiomOfChoice
(fun w : TreeVertex d => IsingInnovation.exists_continuous_posterior
hd hμ hb hcrit uniform01 WienerSampler.sample_isBrownianReal.toIsPreBrownianReal
hmB hcB w)
have hcont (w : TreeVertex d) : ∀ᵐ q ∂μ.prod (labelMeasure d),
Continuous (fun t => Z w t q) := (hZ w).2.2.1
have hver (w : TreeVertex d) (t : ℝ≥0) : Z w t =ᵐ[μ.prod (labelMeasure d)]
(fun q => infiniteGibbsSpinMean beta w
(fun i => (t : ℝ) * spinSign (q.1 i) + B t (q.2 i))) := by
filter_upwards [(hZ w).2.2.2.2.2 t] with q hq
refine hq.trans ?_
change infiniteGibbsSpinMean beta w
(fun i => B t (q.2 i) + (t : ℝ) * spinSign (q.1 i)) = _
apply congrArg (infiniteGibbsSpinMean beta w)
funext i
exact add_comm _ _
have hfields := hμ.continuous_posterior_all_time_field_bound hb
(fun _ : TreeVertex d => uniform01) (fun _ => WienerSampler.sample_isBrownianReal)
(fun _ => hmB) hcont hver
have hc : ∀ᵐ q ∂μ.prod (labelMeasure d),
Continuous (fun t => Real.artanh (Z v t q)) :=
hfields.mono (fun q hq => (hq v).2)
have he (t : ℝ≥0) : (fun q => Real.artanh (Z v t q)) =ᵐ[μ.prod (labelMeasure d)]
(fun q => field beta v t (uniformObservation q)) := by
filter_upwards [hver v t] with q hq
change Real.artanh (Z v t q) = Real.artanh
(infiniteGibbsSpinMean beta v
(fun i => B t (q.2 i) + (t : ℝ) * spinSign (q.1 i)))
apply congrArg Real.artanh
refine hq.trans ?_
apply congrArg (infiniteGibbsSpinMean beta v)
funext i
exact add_comm _ _
exact (ProgressiveVersionPushforward.continuous_progressive_version_of_pullback
measurable_uniformObservation (field_progressive beta v) hc he).2
def continuousDifference {d : ℕ} (beta : ℝ) (v : TreeVertex d)
(t : ℝ≥0) (q : Paths d × (Paths d × Paths d)) : ℝ :=
continuousField beta v t q.2.1 - continuousField beta v t q.2.2
lemma continuousDifference_progressive {d : ℕ} (beta : ℝ) (v : TreeVertex d) :
IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration
(continuousDifference beta v) := by
have hl : IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration
(fun t (q : IsingConditionalCopyBounds.CopySpace d) => field beta v t q.2.1) :=
isStronglyProgressive_artanh _ (isStronglyProgressive_infiniteGibbsSpinMean _ beta v _
IsingConditionalCopyBounds.progressive_leftObservation)
have hr : IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration
(fun t (q : IsingConditionalCopyBounds.CopySpace d) => field beta v t q.2.2) :=
isStronglyProgressive_artanh _ (isStronglyProgressive_infiniteGibbsSpinMean _ beta v _
IsingConditionalCopyBounds.progressive_rightObservation)
exact (ProgressiveVersionPushforward.progressive_version hl).sub
(ProgressiveVersionPushforward.progressive_version hr)
theorem continuousDifference_modification
{d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
[IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (v : TreeVertex d) :
(∀ᵐ q ∂copies beta μ, Continuous (fun t => continuousDifference beta v t q)) ∧
(∀ t, continuousDifference beta v t =ᵐ[copies beta μ]
(fun q => field beta v t q.2.1 - field beta v t q.2.2)) := by
have : IsProbabilityMeasure ((observationLaw μ).map (innovationMap beta)) := inferInstance
obtain ⟨hl, hr⟩ := IsingConditionalCopyBounds.measurePreserving_observation_copies
(beta := beta) (μ := μ) (rho := (observationLaw μ).map (innovationMap beta))
(eta := kernel beta μ) kernel_graphLaw
obtain ⟨hc, he⟩ := continuousField_modification hd hμ hb hcrit v
constructor
· filter_upwards [hl.quasiMeasurePreserving.ae hc, hr.quasiMeasurePreserving.ae hc]
with q hql hqr
exact hql.sub hqr
· intro t
filter_upwards [hl.quasiMeasurePreserving.ae (he t), hr.quasiMeasurePreserving.ae (he t)]
with q hql hqr
exact congrArg₂ (fun x y : ℝ => x - y) hql hqr
end Problem367.IsingObservationFactor
end
end OAI
|