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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 | import OAI.Probability.FreeIsing.Estimates.LocalizedArtanhFormula
import OAI.Probability.FreeIsing.Ising.CanonicalContinuousFields
import OAI.Probability.FreeIsing.Ising.CopyEnergyIntegrability
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal Topology
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation IsingCopyEnergy PosteriorCompensatorStopping
private lemma continuous_real_tanh : Continuous Real.tanh :=
lipschitzWith_tanh_response.continuous
def continuousPosterior {d : ℕ} (beta : ℝ) (v : TreeVertex d)
(t : ℝ≥0) (x : Paths d) : ℝ := Real.tanh (continuousField beta v t x)
lemma continuousPosterior_progressive {d : ℕ} (beta : ℝ) (v : TreeVertex d) :
IsStronglyProgressive (filtration d) (continuousPosterior beta v) := by
intro T
exact continuous_real_tanh.comp_stronglyMeasurable (continuousField_progressive beta v T)
lemma continuousPosterior_interior {d : ℕ} (beta : ℝ) (v : TreeVertex d)
(t : ℝ≥0) (x : Paths d) : continuousPosterior beta v t x ∈ Ioo (-1 : ℝ) 1 :=
⟨Real.neg_one_lt_tanh _, Real.tanh_lt_one _⟩
lemma artanh_continuousPosterior {d : ℕ} (beta : ℝ) (v : TreeVertex d)
(t : ℝ≥0) (x : Paths d) :
Real.artanh (continuousPosterior beta v t x) = continuousField beta v t x :=
Real.artanh_tanh _
theorem continuousPosterior_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 => continuousPosterior beta v t x)) ∧
(∀ t, continuousPosterior beta v t =ᵐ[observationLaw μ] posterior beta v t) := by
obtain ⟨hc, he⟩ := continuousField_modification hd hμ hb hcrit v
refine ⟨hc.mono (fun x hx => continuous_real_tanh.comp hx), fun t => ?_⟩
filter_upwards [he t, tanh_field_eq_posterior_ae hμ hb v t] with x hx ht
exact (congrArg Real.tanh hx).trans ht
lemma copyProjection_measurePreserving
{d : ℕ} (beta : ℝ) (μ : Measure (TreeVertex d → IsingSpin))
[IsProbabilityMeasure μ] (j : Bool) :
MeasurePreserving (projection (d := d) j) (copies beta μ) (observationLaw μ) := 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
cases j
· exact hr
· exact hl
def copyContinuousField {d : ℕ} (beta : ℝ) (v : TreeVertex d) (j : Bool)
(t : ℝ≥0) (q : Joint d) : ℝ := continuousField beta v t (projection j q)
def copyContinuousPosterior {d : ℕ} (beta : ℝ) (v : TreeVertex d) (j : Bool)
(t : ℝ≥0) (q : Joint d) : ℝ := continuousPosterior beta v t (projection j q)
lemma copyContinuousField_progressive {d : ℕ} (beta : ℝ) (v : TreeVertex d) (j : Bool) :
IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration
(copyContinuousField beta v j) := by
cases j
· exact ProgressiveVersionPushforward.progressive_version
(isStronglyProgressive_artanh _ (isStronglyProgressive_infiniteGibbsSpinMean _ beta v _
IsingConditionalCopyBounds.progressive_rightObservation))
· exact ProgressiveVersionPushforward.progressive_version
(isStronglyProgressive_artanh _ (isStronglyProgressive_infiniteGibbsSpinMean _ beta v _
IsingConditionalCopyBounds.progressive_leftObservation))
lemma copyContinuousPosterior_progressive {d : ℕ} (beta : ℝ) (v : TreeVertex d) (j : Bool) :
IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration
(copyContinuousPosterior beta v j) := by
intro T
exact continuous_real_tanh.comp_stronglyMeasurable (copyContinuousField_progressive beta v j T)
theorem copyContinuousPosterior_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) (j : Bool) :
(∀ᵐ q ∂copies beta μ, Continuous (fun t => copyContinuousPosterior beta v j t q)) ∧
(∀ t, copyContinuousPosterior beta v j t =ᵐ[copies beta μ]
(fun q => posterior beta v t (projection j q))) := by
obtain ⟨hc, he⟩ := continuousPosterior_modification hd hμ hb hcrit v
have hp := (copyProjection_measurePreserving beta μ j).quasiMeasurePreserving
exact ⟨hp.ae hc, fun t => hp.ae (he t)⟩
theorem copyContinuousPosterior_compact
{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) (j : Bool) :
∀ᵐ q ∂copies beta μ, ∀ T : ℝ≥0,
∃ r < (1 : ℝ), ∀ t ≤ T, |copyContinuousPosterior beta v j t q| ≤ r := by
filter_upwards [(copyContinuousPosterior_modification hd hμ hb hcrit v j).1] with q hc
intro T
obtain ⟨r, hr0, hr1, hr⟩ := PosteriorInteriorLocalization.exists_uniform_radius_of_compact
(s := Icc (0 : ℝ≥0) T) isCompact_Icc hc.continuousOn
(fun t _ => continuousPosterior_interior beta v t (projection j q))
exact ⟨r, hr1, fun t ht => hr t ⟨zero_le, ht⟩⟩
def copyResponseEnergy {d : ℕ} (beta : ℝ) (v : TreeVertex d) (j : Bool)
(t : ℝ≥0) (q : Joint d) : ℝ :=
infiniteGibbsResponseRow beta v (fun i => projection j q i t)
lemma copyResponseEnergy_progressive {d : ℕ} (beta : ℝ) (v : TreeVertex d) (j : Bool) :
IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration
(copyResponseEnergy beta v j) := by
cases j
· exact isStronglyProgressive_infiniteGibbsResponseRow _ beta v _
IsingConditionalCopyBounds.progressive_rightObservation
· exact isStronglyProgressive_infiniteGibbsResponseRow _ beta v _
IsingConditionalCopyBounds.progressive_leftObservation
theorem copyResponseEnergy_integrable
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
[IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(v : TreeVertex d) (j : Bool) (T : ℝ≥0) :
Integrable (fun p : ℝ × Joint d => copyResponseEnergy beta v j p.1.toNNReal p.2)
((volume.restrict (Ioc (0 : ℝ) T)).prod (copies beta μ)) :=
(copy_row_energy_integrable hμ hb v j T.coe_nonneg).2.1
def commonResponseCompensator {d : ℕ} (beta : ℝ) (v : TreeVertex d) :
ℝ≥0 → Joint d → ℝ :=
nnrealTimePrimitive (fun t q =>
copyResponseEnergy beta v true t q + copyResponseEnergy beta v false t q)
lemma commonResponseCompensator_progressive {d : ℕ} (beta : ℝ) (v : TreeVertex d) :
IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration
(commonResponseCompensator beta v) :=
isStronglyProgressive_nnrealTimePrimitive
((copyResponseEnergy_progressive beta v true).add (copyResponseEnergy_progressive beta v false))
theorem commonResponseCompensator_continuous
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
[IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (v : TreeVertex d) :
∀ᵐ q ∂copies beta μ, Continuous (fun t => commonResponseCompensator beta v t q) := by
apply ae_continuous_nnrealTimePrimitive_of_joint
intro n
have hi := (copyResponseEnergy_integrable hμ hb v true (n : ℝ≥0)).add
(copyResponseEnergy_integrable hμ hb v false (n : ℝ≥0))
convert hi using 1; rfl
def commonPosteriorExit {d : ℕ} (beta : ℝ) (v : TreeVertex d)
(n : ℕ) (q : Joint d) : WithTop ℝ≥0 :=
min (jointExit (copyContinuousPosterior beta v true) (commonResponseCompensator beta v) n q)
(jointExit (copyContinuousPosterior beta v false) (commonResponseCompensator beta v) n q)
theorem isLocalizingSequence_commonPosteriorExit
{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) :
IsLocalizingSequence ContinuousCopyBrownian.joinedFiltration.rightCont
(commonPosteriorExit beta v) (copies beta μ) := by
have h (j : Bool) := isLocalizingSequence_jointExit_rightCont
(copyContinuousPosterior_progressive beta v j).stronglyAdapted.adapted
(commonResponseCompensator_progressive beta v).stronglyAdapted.adapted
(copyContinuousPosterior_compact hd hμ hb hcrit v j)
(commonResponseCompensator_continuous hμ hb v)
exact (h true).min (h false)
theorem ae_eventually_le_commonPosteriorExit
{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)
(T : ℝ≥0) :
∀ᵐ q ∂copies beta μ, ∀ᶠ n in atTop,
(T : WithTop ℝ≥0) ≤ commonPosteriorExit beta v n q := by
filter_upwards [copyContinuousPosterior_compact hd hμ hb hcrit v true,
copyContinuousPosterior_compact hd hμ hb hcrit v false,
commonResponseCompensator_continuous hμ hb v] with q hl hr hA
filter_upwards [eventually_le_jointExit (hl T) hA,
eventually_le_jointExit (hr T) hA] with n hn hn'
exact le_min hn hn'
end Problem367.IsingObservationFactor
end
end OAI
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