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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 | import OAI.Probability.FreeIsing.Energy.DiscrepancyRowEnergy
import OAI.Probability.FreeIsing.Energy.NormalizedResponseLimit
import OAI.Probability.FreeIsing.Posterior.InfinitePosteriorField
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology ENNReal NNReal
namespace Problem367
variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
theorem integrable_sq_and_integral_le_of_tendsto_ae
(μ : Measure Ω) {X : ℕ → Ω → ℝ} {Y : Ω → ℝ} {K : ℝ}
(hX : ∀ n, AEStronglyMeasurable (X n) μ)
(hY : AEStronglyMeasurable Y μ) (hK : 0 ≤ K)
(hlim : ∀ᵐ ω ∂μ, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω)))
(hbound : ∀ᶠ n in atTop, Integrable (fun ω => (X n ω) ^ 2) μ ∧
(∫ ω, (X n ω) ^ 2 ∂μ) ≤ K) :
Integrable (fun ω => (Y ω) ^ 2) μ ∧ (∫ ω, (Y ω) ^ 2 ∂μ) ≤ K := by
have hfatou : (∫⁻ ω, ENNReal.ofReal ((Y ω) ^ 2) ∂μ) ≤ ENNReal.ofReal K := by
calc
(∫⁻ ω, ENNReal.ofReal ((Y ω) ^ 2) ∂μ) =
∫⁻ ω, liminf (fun n => ENNReal.ofReal ((X n ω) ^ 2)) atTop ∂μ := by
apply lintegral_congr_ae
filter_upwards [hlim] with ω hω
exact ((ENNReal.continuous_ofReal.tendsto _).comp (hω.pow 2)).liminf_eq.symm
_ ≤ liminf (fun n => ∫⁻ ω, ENNReal.ofReal ((X n ω) ^ 2) ∂μ) atTop :=
lintegral_liminf_le' (fun n => ((hX n).pow 2).aemeasurable.ennreal_ofReal)
_ ≤ ENNReal.ofReal K := by
apply liminf_le_of_frequently_le'
apply Filter.Eventually.frequently
filter_upwards [hbound] with n hn
rw [← ofReal_integral_eq_lintegral_ofReal hn.1
(Filter.Eventually.of_forall (fun ω => sq_nonneg (X n ω)))]
exact ENNReal.ofReal_le_ofReal hn.2
refine ⟨⟨hY.pow 2, ?_⟩, ?_⟩
· apply (hasFiniteIntegral_iff_ofReal
(Filter.Eventually.of_forall (fun ω => sq_nonneg (Y ω)))).mpr
exact hfatou.trans_lt ENNReal.ofReal_lt_top
· rw [integral_eq_lintegral_of_nonneg_ae
(Filter.Eventually.of_forall (fun ω => sq_nonneg (Y ω))) (hY.pow 2)]
exact ENNReal.toReal_le_of_le_ofReal hK hfatou
theorem posteriorResponse_integrable_sq_of_uniform_bound
(μ : Measure Ω) [IsFiniteMeasure μ] (F : Filtration ℕ mΩ)
(f g : Ω → ℝ) {K : ℝ} (hK : 0 ≤ K)
(hinterior : ∀ᵐ ω ∂μ, μ[f | ⨆ n, F n] ω ∈ Set.Ioo (-1 : ℝ) 1)
(hbound : ∀ᶠ n in atTop,
Integrable (fun ω => (posteriorResponse μ (F n) f g ω) ^ 2) μ ∧
(∫ ω, (posteriorResponse μ (F n) f g ω) ^ 2 ∂μ) ≤ K) :
Integrable (fun ω => (posteriorResponse μ (⨆ n, F n) f g ω) ^ 2) μ ∧
(∫ ω, (posteriorResponse μ (⨆ n, F n) f g ω) ^ 2 ∂μ) ≤ K := by
exact integrable_sq_and_integral_le_of_tendsto_ae μ
(fun n => ((stronglyMeasurable_posteriorResponse μ (F n) f g).mono
(F.le n)).aestronglyMeasurable)
(((stronglyMeasurable_posteriorResponse μ (⨆ n, F n) f g).mono
(iSup_le fun n => F.le n)).aestronglyMeasurable)
hK (tendsto_ae_posteriorResponse μ F f g hinterior) hbound
theorem posteriorResponse_row_energy_of_uniform_bounds
{ι : Type*} [Countable ι]
(μ : Measure Ω) [IsFiniteMeasure μ] (F : Filtration ℕ mΩ)
(f : Ω → ℝ) (g : ι → Ω → ℝ) (K : ι → ℝ)
(hK : ∀ i, 0 ≤ K i) (hKs : Summable K)
(hinterior : ∀ᵐ ω ∂μ, μ[f | ⨆ n, F n] ω ∈ Set.Ioo (-1 : ℝ) 1)
(hbound : ∀ i, ∀ᶠ n in atTop,
Integrable (fun ω => (posteriorResponse μ (F n) f (g i) ω) ^ 2) μ ∧
(∫ ω, (posteriorResponse μ (F n) f (g i) ω) ^ 2 ∂μ) ≤ K i) :
(∀ᵐ ω ∂μ, Summable (fun i => (posteriorResponse μ (⨆ n, F n) f (g i) ω) ^ 2)) ∧
Integrable (fun ω => ∑' i, (posteriorResponse μ (⨆ n, F n) f (g i) ω) ^ 2) μ ∧
(∫ ω, (∑' i, (posteriorResponse μ (⨆ n, F n) f (g i) ω) ^ 2) ∂μ) ≤
∑' i, K i := by
have hb (i : ι) := posteriorResponse_integrable_sq_of_uniform_bound μ F f (g i)
(hK i) hinterior (hbound i)
have hs : Summable (fun i =>
∫ ω, (posteriorResponse μ (⨆ n, F n) f (g i) ω) ^ 2 ∂μ) :=
Summable.of_nonneg_of_le (fun i => integral_nonneg (fun ω => sq_nonneg _))
(fun i => (hb i).2) hKs
obtain ⟨hae, hint⟩ := DiscrepancyEnergy.sq_row_integrable_of_summable_integrals
(fun i => (hb i).1) hs
refine ⟨hae, hint, ?_⟩
have heq := DiscrepancyEnergy.hasSum_integral_sq_row
(fun i => ((stronglyMeasurable_posteriorResponse μ (⨆ n, F n) f (g i)).mono
(iSup_le fun n => F.le n)).aestronglyMeasurable) hae hint
rw [← heq.tsum_eq]
exact hs.tsum_le_tsum (fun i => (hb i).2) hKs
section Gaussian
open PlantedSymmetry
def gaussianPosteriorResponse {d : ℕ}
(μ : Measure (TreeVertex d → IsingSpin)) (t : ℝ≥0)
(v w : TreeVertex d) :
((TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) → ℝ :=
posteriorResponse (μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(observationSigma (TreeVertex d) t)
(fun p => spinSign (p.1 v)) (fun p => spinSign (p.1 w))
theorem HasFreeIsingLaw.finiteObservation_posteriorResponse
{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)
(v w : D) :
posteriorResponse (μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(MeasurableSpace.comap (finiteObservation D t) inferInstance)
(fun p => spinSign (p.1 v.1)) (fun p => spinSign (p.1 w.1)) =ᵐ[
μ.prod (gaussianNoiseLaw (TreeVertex d) t)]
fun p =>
let M := fun q => FinitePosterior.fieldMean (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) u => spinSign (eta u)) q (finiteObservation D t p)
(M (fun eta => spinSign (eta v) * spinSign (eta w)) -
M (fun eta => spinSign (eta v)) * M (fun eta => spinSign (eta w))) /
(1 - (M (fun eta => spinSign (eta v))) ^ 2) := by
filter_upwards [hμ.finiteObservation_condExp D hD t
(fun eta => spinSign (eta v) * spinSign (eta w)),
hμ.finiteObservation_condExp D hD t (fun eta => spinSign (eta v)),
hμ.finiteObservation_condExp D hD t (fun eta => spinSign (eta w))]
with p hpair hv hw
dsimp only [posteriorResponse, posteriorCovariance]
simp only [Finset.restrict] at hpair hv hw
rw [hpair, hv, hw]
theorem HasFreeIsingLaw.gaussian_posteriorResponse_tendsto_ae
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(D : ℕ → Finset (TreeVertex d)) (hD : Monotone D)
(hex : ∀ w, ∃ n, w ∈ D n) (t : ℝ≥0) (v w : TreeVertex d) :
∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t),
Tendsto (fun n => posteriorResponse (μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(MeasurableSpace.comap (finiteObservation (D n) t) inferInstance)
(fun p => spinSign (p.1 v)) (fun p => spinSign (p.1 w)) p) atTop
(𝓝 (gaussianPosteriorResponse μ t v w p)) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have hi := (hμ.gaussian_posterior_field_bound hbeta t v).mono (fun _ h => h.1)
have hi' : ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t),
(μ.prod (gaussianNoiseLaw (TreeVertex d) t))[
fun p => spinSign (p.1 v) | ⨆ n, finiteObservationFiltration D hD t n] p ∈
Set.Ioo (-1 : ℝ) 1 := by
rw [finiteObservationFiltration_iSup D hD hex t]
exact hi
have h := tendsto_ae_posteriorResponse
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) (finiteObservationFiltration D hD t)
(fun p => spinSign (p.1 v)) (fun p => spinSign (p.1 w)) hi'
rw [finiteObservationFiltration_iSup D hD hex t] at h
exact h
theorem HasFreeIsingLaw.gaussian_posteriorResponse_integrable_sq
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(D : ℕ → Finset (TreeVertex d)) (hD : Monotone D)
(hex : ∀ w, ∃ n, w ∈ D n) (t : ℝ≥0) (v w : TreeVertex d)
{K : ℝ} (hK : 0 ≤ K)
(hbound : ∀ᶠ n in atTop,
Integrable (fun p => (posteriorResponse (μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(MeasurableSpace.comap (finiteObservation (D n) t) inferInstance)
(fun p => spinSign (p.1 v)) (fun p => spinSign (p.1 w)) p) ^ 2)
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ∧
(∫ p, (posteriorResponse (μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(MeasurableSpace.comap (finiteObservation (D n) t) inferInstance)
(fun p => spinSign (p.1 v)) (fun p => spinSign (p.1 w)) p) ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ K) :
Integrable (fun p => (gaussianPosteriorResponse μ t v w p) ^ 2)
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ∧
(∫ p, (gaussianPosteriorResponse μ t v w p) ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ K := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
apply integrable_sq_and_integral_le_of_tendsto_ae
(μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(fun n => ((stronglyMeasurable_posteriorResponse _ _ _ _).mono
(measurable_finiteObservation (D n) t).comap_le).aestronglyMeasurable)
(((stronglyMeasurable_posteriorResponse _ _ _ _).mono
(observationSigma_le (TreeVertex d) t)).aestronglyMeasurable)
hK (hμ.gaussian_posteriorResponse_tendsto_ae hbeta D hD hex t v w) hbound
theorem HasFreeIsingLaw.gaussian_posteriorResponse_meanSquare
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(D : ℕ → Finset (TreeVertex d)) (hD : Monotone D)
(hex : ∀ w, ∃ n, w ∈ D n) (t : ℝ≥0) (v w : TreeVertex d) :
Tendsto (fun n => ∫ p,
(posteriorResponse (μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(MeasurableSpace.comap (finiteObservation (D n) t) inferInstance)
(fun p => spinSign (p.1 v)) (fun p => spinSign (p.1 w)) p -
gaussianPosteriorResponse μ t v w p) ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) atTop (𝓝 0) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have hi := (hμ.gaussian_posterior_field_bound hbeta t v).mono (fun _ h => h.1)
have hi' : ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t),
(μ.prod (gaussianNoiseLaw (TreeVertex d) t))[
fun p => spinSign (p.1 v) | ⨆ n, finiteObservationFiltration D hD t n] p ∈
Set.Ioo (-1 : ℝ) 1 := by
rw [finiteObservationFiltration_iSup D hD hex t]
exact hi
have hb : ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t),
spinSign (p.1 v) = -1 ∨ spinSign (p.1 v) = 1 :=
Filter.Eventually.of_forall (fun p => by cases p.1 v <;> simp [spinSign])
have hgb : ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t),
|spinSign (p.1 w)| ≤ 1 :=
Filter.Eventually.of_forall (fun p => by cases p.1 w <;> norm_num [spinSign])
have h := tendsto_integral_sq_sub_posteriorResponse
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) (finiteObservationFiltration D hD t)
(integrable_plantedSpin _ v) (integrable_plantedSpin _ w) hb hgb hi'
rw [finiteObservationFiltration_iSup D hD hex t] at h
exact h
end Gaussian
end Problem367
end
end OAI
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