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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 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 | import OAI.Probability.FreeIsing.Gaussian.FiniteGaussianPlanted
import OAI.Probability.FreeIsing.Ising.GaussianPosterior
import OAI.Probability.FreeIsing.Posterior.Limit
import OAI.Probability.FreeIsing.Posterior.Symmetry
import OAI.Probability.FreeIsing.Trees.ConnectedExhaustion
import OAI.Probability.FreeIsing.Trees.GibbsField
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal Topology
namespace Problem367
open PlantedSymmetry
local instance infinitePosteriorFieldNonemptySpin : Nonempty IsingSpin := ⟨.neg⟩
def finiteObservation {d : ℕ} (D : Finset (TreeVertex d)) (t : ℝ≥0)
(p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) : D → ℝ :=
D.restrict (observation t p)
theorem measurable_finiteObservation {d : ℕ} (D : Finset (TreeVertex d)) (t : ℝ≥0) :
Measurable (finiteObservation D t) :=
D.measurable_restrict.comp (measurable_observation t)
theorem gaussianPlanted_restrict_joint_law {d : ℕ}
(μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ]
(D : Finset (TreeVertex d)) (t : ℝ≥0) :
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)).map
(fun p => (D.restrict p.1, finiteObservation D t p)) =
(μ.map D.restrict) ⊗ₘ finiteGaussianObservationKernel t
(fun (eta : D → IsingSpin) v => spinSign (eta v)) := by
let R : ((TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) →
(D → IsingSpin) × (D → ℝ) := Prod.map D.restrict D.restrict
have hR : Measurable R := D.measurable_restrict.prodMap D.measurable_restrict
have hRlaw : (μ.prod (gaussianNoiseLaw (TreeVertex d) t)).map R =
(μ.map D.restrict).prod (Measure.pi (fun _ : D => gaussianReal 0 t)) := by
simpa only [R, gaussianNoiseLaw, Measure.infinitePi_map_restrict] using
(Measure.map_prod_map μ (gaussianNoiseLaw (TreeVertex d) t)
D.measurable_restrict D.measurable_restrict).symm
let F : (D → IsingSpin) × (D → ℝ) → (D → IsingSpin) × (D → ℝ) :=
fun p => (p.1, fun v => (t : ℝ) * spinSign (p.1 v) + p.2 v)
have hF : Measurable F := by
apply measurable_fst.prodMk
apply Measurable.of_eval
intro v
exact (measurable_const.mul
((measurable_of_finite spinSign).comp ((measurable_pi_apply v).comp measurable_fst))).add
((measurable_pi_apply v).comp measurable_snd)
have : IsProbabilityMeasure (μ.map D.restrict) := inferInstance
calc
_ = ((μ.prod (gaussianNoiseLaw (TreeVertex d) t)).map R).map F := by
rw [Measure.map_map hF hR]
rfl
_ = _ := by
rw [hRlaw]
exact finiteGaussianPlanted_joint_law (μ.map D.restrict) t
(fun (eta : D → IsingSpin) v => spinSign (eta v))
theorem HasFreeIsingLaw.finiteObservation_condExp {d : ℕ} {beta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ)
(D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (t : ℝ≥0)
(q : (D → IsingSpin) → ℝ) :
(μ.prod (gaussianNoiseLaw (TreeVertex d) t))[
fun p => q (D.restrict p.1) |
MeasurableSpace.comap (finiteObservation D t) inferInstance] =ᵐ[
μ.prod (gaussianNoiseLaw (TreeVertex d) t)]
fun p => FinitePosterior.fieldMean (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) v => spinSign (eta v)) q (finiteObservation D t p) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
let : IsProbabilityMeasure (μ.map D.restrict) := inferInstance
exact FiniteBayes.condExp_state_eq_of_posterior_integral
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) (μ.map D.restrict)
(finiteGaussianObservationKernel t (fun (eta : D → IsingSpin) v => spinSign (eta v)))
(fun p => D.restrict p.1) (finiteObservation D t)
(D.measurable_restrict.comp measurable_fst) (measurable_finiteObservation D t)
(gaussianPlanted_restrict_joint_law μ D t) q _
(hμ.gaussian_restrict_posterior_mean D hD t q)
theorem finiteIsingPosterior_tanh_bounds {d : ℕ} {beta : ℝ}
(hbeta : 0 ≤ beta) (D : Finset (TreeVertex d)) (y : D → ℝ) (v : D) :
Real.tanh (y v - (d : ℝ) * beta) ≤
FinitePosterior.fieldMean (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) w => spinSign (eta w))
(fun eta => spinSign (eta v)) y ∧
FinitePosterior.fieldMean (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) w => spinSign (eta w))
(fun eta => spinSign (eta v)) y ≤ Real.tanh (y v + (d : ℝ) * beta) := by
have hmem := FinitePosterior.fieldMean_spin_mem_Ioo (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) w => spinSign (eta w))
(fun eta => spinSign (eta v)) y
(fun eta => by cases eta v <;> simp [spinSign])
⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩
have hb := finite_tree_gibbs_field_uniform_bound D hbeta y v
change |Real.artanh (FinitePosterior.fieldMean (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) w => spinSign (eta w))
(fun eta => spinSign (eta v)) y) - y v| ≤ (d : ℝ) * beta at hb
have hmono : Monotone Real.tanh := by
apply monotone_of_deriv_nonneg
· intro x
exact (hasDerivAt_tanh_response x).differentiableAt
· intro x
rw [(hasDerivAt_tanh_response x).deriv]
exact (sub_pos.mpr (Real.tanh_sq_lt_one x)).le
have hlo := hmono (show y v - (d : ℝ) * beta ≤
Real.artanh (FinitePosterior.fieldMean (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) w => spinSign (eta w))
(fun eta => spinSign (eta v)) y) by linarith [(abs_le.mp hb).1])
have hhi := hmono (show Real.artanh (FinitePosterior.fieldMean (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) w => spinSign (eta w))
(fun eta => spinSign (eta v)) y) ≤ y v + (d : ℝ) * beta by
linarith [(abs_le.mp hb).2])
rw [Real.tanh_artanh hmem] at hlo hhi
exact ⟨hlo, hhi⟩
theorem HasFreeIsingLaw.finiteObservation_spin_tanh_bounds {d : ℕ} {beta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (t : ℝ≥0) (v : D) :
∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t),
Real.tanh (observation t p v.1 - (d : ℝ) * beta) ≤
(μ.prod (gaussianNoiseLaw (TreeVertex d) t))[
fun p => spinSign (p.1 v.1) |
MeasurableSpace.comap (finiteObservation D t) inferInstance] p ∧
(μ.prod (gaussianNoiseLaw (TreeVertex d) t))[
fun p => spinSign (p.1 v.1) |
MeasurableSpace.comap (finiteObservation D t) inferInstance] p ≤
Real.tanh (observation t p v.1 + (d : ℝ) * beta) := by
filter_upwards [hμ.finiteObservation_condExp D hD t (fun eta => spinSign (eta v))]
with p hp
change (μ.prod (gaussianNoiseLaw (TreeVertex d) t))[
fun p => spinSign (p.1 v.1) |
MeasurableSpace.comap (finiteObservation D t) inferInstance] p = _ at hp
rw [hp]
exact finiteIsingPosterior_tanh_bounds hbeta D (finiteObservation D t p) v
def finiteObservationFiltration {d : ℕ} (D : ℕ → Finset (TreeVertex d))
(hD : Monotone D) (t : ℝ≥0) :
Filtration ℕ (inferInstance : MeasurableSpace
((TreeVertex d → IsingSpin) × (TreeVertex d → ℝ))) where
seq n := MeasurableSpace.comap (finiteObservation (D n) t) inferInstance
mono' n m h := by
change MeasurableSpace.comap (fun p (v : D n) => observation t p v.1) inferInstance ≤
MeasurableSpace.comap (fun p (v : D m) => observation t p v.1) inferInstance
rw [MeasurableSpace.comap_process_pi, MeasurableSpace.comap_process_pi]
apply iSup_le
intro v
exact le_iSup_of_le (⟨v.1, hD h v.2⟩ : D m) le_rfl
le' n := (measurable_finiteObservation (D n) t).comap_le
theorem finiteObservationFiltration_iSup {d : ℕ}
(D : ℕ → Finset (TreeVertex d)) (hD : Monotone D)
(hex : ∀ v, ∃ n, v ∈ D n) (t : ℝ≥0) :
(⨆ n, finiteObservationFiltration D hD t n) = observationSigma (TreeVertex d) t := by
change (⨆ n, MeasurableSpace.comap
(fun p (v : D n) => observation t p v.1) inferInstance) =
MeasurableSpace.comap (fun p (v : TreeVertex d) => observation t p v) inferInstance
simp only [MeasurableSpace.pi, MeasurableSpace.comap_iSup,
MeasurableSpace.comap_comp, Function.comp_def]
apply le_antisymm
· apply iSup_le
intro n
apply iSup_le
intro v
exact le_iSup_of_le v.1 le_rfl
· apply iSup_le
intro v
obtain ⟨n, hn⟩ := hex v
exact le_iSup_of_le n (le_iSup_of_le (⟨v, hn⟩ : D n) le_rfl)
theorem HasFreeIsingLaw.posterior_field_bound_of_exhaustion {d : ℕ} {beta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(v : TreeVertex d) (D : ℕ → Finset (TreeVertex d)) (hD : Monotone D)
(hconn : ∀ n, ConnectedFinset (D n)) (hv : ∀ n, v ∈ D n)
(hex : ∀ w, ∃ n, w ∈ D n) (t : ℝ≥0) :
∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t),
posteriorMagnetization (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) t v p ∈
Set.Ioo (-1 : ℝ) 1 ∧
|Real.artanh (posteriorMagnetization
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) t v p) - observation t p v| ≤
(d : ℝ) * beta := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have hb (n : ℕ) := hμ.finiteObservation_spin_tanh_bounds hbeta
(D n) (hconn n) t ⟨v, hv n⟩
have hlim := tendsto_posterior_fields_ae
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) (finiteObservationFiltration D hD t)
(fun p => spinSign (p.1 v)) (fun p => observation t p v) ((d : ℝ) * beta) hb
rw [finiteObservationFiltration_iSup D hD hex t] at hlim
filter_upwards [hlim] with p hp
exact ⟨hp.1, hp.2.1⟩
theorem HasFreeIsingLaw.gaussian_posterior_field_bound {d : ℕ} {beta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (v : TreeVertex d) :
∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t),
posteriorMagnetization (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) t v p ∈
Set.Ioo (-1 : ℝ) 1 ∧
|Real.artanh (posteriorMagnetization
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) t v p) - observation t p v| ≤
(d : ℝ) * beta := by
obtain ⟨D, hD, hconn, hex⟩ := exists_connected_exhaustion v
exact hμ.posterior_field_bound_of_exhaustion hbeta v D hD
(fun n => (hconn n).1) (fun n => (hconn n).2) hex t
end Problem367
end
end OAI
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