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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 | import OAI.Probability.FreeIsing.Estimates.DiscrepancyDensityModification
import OAI.Probability.FreeIsing.Ising.CanonicalPosteriorStops
import OAI.Probability.FreeIsing.Ising.ContinuousRootResponse
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.IsingContinuousCopyDensity
open IsingObservationInnovation IsingObservationFactor IsingCopyEnergy
variable {d : ℕ} {beta : ℝ}
def response (hb : 0 ≤ beta) (v : TreeVertex d) (j : Bool)
(t : ℝ≥0) (q : Joint d) : ℝ :=
continuousRootResponse hb v t (projection j q)
theorem response_progressive (hb : 0 ≤ beta) (v : TreeVertex d) (j : Bool) :
IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration (response hb v j) := by
intro T
let : MeasurableSpace (Joint d) := ContinuousCopyBrownian.joinedFiltration T
exact ((continuous_siteRootResponse hb v).measurable.comp
(Measurable.of_eval (fun i =>
(copyContinuousField_progressive beta i j T).measurable))).stronglyMeasurable
theorem response_eq_raw_ae
(hd : 3 ≤ d) {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) (t : ℝ≥0) :
response hb v j t =ᵐ[copies beta μ]
fun q => responseRow beta (ConnectedHull.root d) v ((t : ℝ), projection j q) := by
have h := (copyProjection_measurePreserving beta μ j).quasiMeasurePreserving.ae
(continuousRootResponse_eq_response_ae hd hμ hb hcrit v t)
filter_upwards [h] with q hq
simpa only [response, responseRow, Real.toNNReal_coe] using hq
private theorem rawObservation_progressive (j : Bool) (v : TreeVertex d) :
IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration
(fun t (q : Joint d) => projection j q v t) := by
cases j
· exact IsingConditionalCopyBounds.progressive_rightObservation v
· exact IsingConditionalCopyBounds.progressive_leftObservation v
private theorem rawPosterior_progressive (v : TreeVertex d) (j : Bool) :
IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration
(fun t (q : Joint d) => posterior beta v t (projection j q)) :=
isStronglyProgressive_infiniteGibbsSpinMean _ beta v _ (rawObservation_progressive j)
private theorem rawResponse_progressive (v : TreeVertex d) (j : Bool) :
IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration
(fun t (q : Joint d) =>
responseRow beta (ConnectedHull.root d) v ((t : ℝ), projection j q)) := by
intro T
let : MeasurableSpace (Joint d) := ContinuousCopyBrownian.joinedFiltration T
simpa only [Function.comp_def, responseRow, Real.toNNReal_coe] using
((measurable_infiniteGibbsResponse beta (ConnectedHull.root d) v).comp
(Measurable.of_eval (fun i =>
(rawObservation_progressive j i T).measurable))).stronglyMeasurable
def continuousDensity (hb : 0 ≤ beta) (t : ℝ≥0) : Joint d → ℝ :=
DiscrepancyEnergy.density (fun v => response hb v true t)
(fun v => response hb v false t) (continuousDifference beta (ConnectedHull.root d) t)
(copyContinuousPosterior beta (ConnectedHull.root d) true t)
(copyContinuousPosterior beta (ConnectedHull.root d) false t)
theorem continuousDensity_progressive (hb : 0 ≤ beta) :
IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration
(continuousDensity (d := d) hb) :=
DiscrepancyEnergy.isStronglyProgressive_density (ι := TreeVertex d) (Ω := Joint d)
(fun v => response_progressive hb v true) (fun v => response_progressive hb v false)
(continuousDifference_progressive beta (ConnectedHull.root d))
(copyContinuousPosterior_progressive beta (ConnectedHull.root d) true)
(copyContinuousPosterior_progressive beta (ConnectedHull.root d) false)
theorem actualDensity_progressive :
IsStronglyProgressive ContinuousCopyBrownian.joinedFiltration
(fun t : ℝ≥0 => IsingCopyEnergy.density beta (ConnectedHull.root d) (t : ℝ)) := by
unfold IsingCopyEnergy.density
simp only [Real.toNNReal_coe]
apply DiscrepancyEnergy.isStronglyProgressive_density
· intro v
exact rawResponse_progressive (beta := beta) v true
· intro v
exact rawResponse_progressive (beta := beta) v false
· exact (isStronglyProgressive_artanh _
(rawPosterior_progressive (beta := beta) (ConnectedHull.root d) true)).sub
(isStronglyProgressive_artanh _
(rawPosterior_progressive (beta := beta) (ConnectedHull.root d) false))
· exact rawPosterior_progressive (beta := beta) (ConnectedHull.root d) true
· exact rawPosterior_progressive (beta := beta) (ConnectedHull.root d) false
theorem continuousDensity_eq_actual_ae
(hd : 3 ≤ d) {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) (t : ℝ≥0) :
continuousDensity hb t =ᵐ[copies beta μ]
IsingCopyEnergy.density beta (ConnectedHull.root d) (t : ℝ) := by
unfold continuousDensity IsingCopyEnergy.density
simp only [Real.toNNReal_coe]
apply DiscrepancyEnergy.density_congr_ae
· intro v
exact response_eq_raw_ae hd hμ hb hcrit v true t
· intro v
exact response_eq_raw_ae hd hμ hb hcrit v false t
· exact (continuousDifference_modification hd hμ hb hcrit (ConnectedHull.root d)).2 t
· exact (copyContinuousPosterior_modification hd hμ hb hcrit
(ConnectedHull.root d) true).2 t
· exact (copyContinuousPosterior_modification hd hμ hb hcrit
(ConnectedHull.root d) false).2 t
theorem continuousDensity_eq_actual_prod_ae
(hd : 3 ≤ d) {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) (T : ℝ≥0) :
(fun p : ℝ × Joint d => continuousDensity hb p.1.toNNReal p.2) =ᵐ[
(volume.restrict (Set.Ioc (0 : ℝ) (T : ℝ))).prod (copies beta μ)]
(fun p => IsingCopyEnergy.density beta (ConnectedHull.root d) p.1 p.2) := by
have hmap : Measurable (fun p : ℝ × Joint d => (p.1.toNNReal, p.2)) :=
(measurable_real_toNNReal.comp measurable_fst).prodMk measurable_snd
have hl := (stronglyMeasurable_joint_of_isStronglyProgressive
(continuousDensity_progressive hb)).comp_measurable hmap
have hr := (stronglyMeasurable_joint_of_isStronglyProgressive
(actualDensity_progressive (d := d) (beta := beta))).comp_measurable hmap
have he : (fun p : ℝ × Joint d => continuousDensity hb p.1.toNNReal p.2) =ᵐ[
(volume.restrict (Set.Ioc (0 : ℝ) (T : ℝ))).prod (copies beta μ)]
(fun p => IsingCopyEnergy.density beta (ConnectedHull.root d)
(p.1.toNNReal : ℝ) p.2) := by
apply (Measure.ae_prod_iff_ae_ae (measurableSet_eq_fun hl.measurable hr.measurable)).2
exact Filter.Eventually.of_forall fun t =>
continuousDensity_eq_actual_ae hd hμ hb hcrit t.toNNReal
simpa only [IsingCopyEnergy.density, responseRow, Real.toNNReal_coe] using he
theorem continuousDensity_integrable
(hd : 3 ≤ d) {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) (T : ℝ≥0) :
Integrable (fun p : ℝ × Joint d => continuousDensity hb p.1.toNNReal p.2)
((volume.restrict (Set.Ioc (0 : ℝ) (T : ℝ))).prod (copies beta μ)) :=
(IsingCopyEnergy.density_integrable hμ hb (ConnectedHull.root d) T).congr
(continuousDensity_eq_actual_prod_ae hd hμ hb hcrit T).symm
theorem ae_all_primitive_eq_actual
(hd : 3 ≤ d) {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) :
∀ᵐ q ∂copies beta μ, ∀ t,
nnrealTimePrimitive (continuousDensity hb) t q =
nnrealTimePrimitive
(fun s : ℝ≥0 => IsingCopyEnergy.density beta (ConnectedHull.root d) (s : ℝ)) t q :=
ae_all_nnrealTimePrimitive_eq_of_modification (continuousDensity_progressive hb)
actualDensity_progressive (continuousDensity_eq_actual_ae hd hμ hb hcrit)
theorem ae_all_stopped_primitive_eq_actual
(hd : 3 ≤ d) {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)
(τ : Joint d → WithTop ℝ≥0) :
∀ᵐ q ∂copies beta μ, ∀ t,
stoppedProcess (nnrealTimePrimitive (continuousDensity hb)) τ t q =
stoppedProcess (nnrealTimePrimitive
(fun s : ℝ≥0 => IsingCopyEnergy.density beta (ConnectedHull.root d) (s : ℝ))) τ t q := by
filter_upwards [ae_all_primitive_eq_actual hd hμ hb hcrit] with q hq
intro t
exact hq _
theorem integral_cutoff_density_eq_actual
(hd : 3 ≤ d) {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)
(τ : Joint d → WithTop ℝ≥0) (t : ℝ≥0) :
(∫ q, if (t : WithTop ℝ≥0) ≤ τ q then continuousDensity hb t q else 0
∂copies beta μ) =
∫ q, if (t : WithTop ℝ≥0) ≤ τ q then
IsingCopyEnergy.density beta (ConnectedHull.root d) (t : ℝ) q else 0
∂copies beta μ := by
apply integral_congr_ae
filter_upwards [continuousDensity_eq_actual_ae hd hμ hb hcrit t] with q hq
by_cases ht : (t : WithTop ℝ≥0) ≤ τ q
· simp only [ht, ite_true, hq]
· simp only [ht, ite_false]
theorem integral_stopDensity_eq_actual
(hd : 3 ≤ d) {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)
(τ : Joint d → WithTop ℝ≥0) (s : ℝ) :
(∫ q, StoppedTimePrimitive.stopDensity (continuousDensity hb) τ s.toNNReal q
∂copies beta μ) =
∫ q, if (s.toNNReal : WithTop ℝ≥0) ≤ τ q then
IsingCopyEnergy.density beta (ConnectedHull.root d) s q else 0
∂copies beta μ := by
simpa only [StoppedTimePrimitive.stopDensity, IsingCopyEnergy.density,
responseRow, Real.toNNReal_coe] using
integral_cutoff_density_eq_actual hd hμ hb hcrit τ s.toNNReal
theorem integral_integral_stopDensity_eq_actual
(hd : 3 ≤ d) {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)
(τ : Joint d → WithTop ℝ≥0) (T : ℝ≥0) :
(∫ s in (0 : ℝ)..(T : ℝ), ∫ q,
StoppedTimePrimitive.stopDensity (continuousDensity hb) τ s.toNNReal q
∂copies beta μ) =
∫ s in (0 : ℝ)..(T : ℝ), ∫ q,
if (s.toNNReal : WithTop ℝ≥0) ≤ τ q then
IsingCopyEnergy.density beta (ConnectedHull.root d) s q else 0
∂copies beta μ := by
apply intervalIntegral.integral_congr
intro s _
exact integral_stopDensity_eq_actual hd hμ hb hcrit τ s
end Problem367.IsingContinuousCopyDensity
end
end OAI
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