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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 | import OAI.Probability.FreeIsing.Gaussian.BinaryPosteriorResponseBound
import OAI.Probability.FreeIsing.Posterior.NormalizedPosteriorRowConvergence
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.NormalizedResponseTime
open PlantedSymmetry GaussianTreeResponse
lemma observedRow_abs_le_one {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta)
(D : Finset (TreeVertex d)) (hD : ConnectedFinset D)
(v : TreeVertex d) (hv : v ∈ D) (t : ℝ) (w : TreeVertex d)
(p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) :
|observedRow beta D v t w p| ≤ 1 := by
classical
by_cases hw : w ∈ D
· simp only [observedRow, ite_eq_left hw]
have hb := TreePathResponse.response_mem_Icc hbeta (observation t p) hD
(⟨v, hv⟩ : D) (⟨w, hw⟩ : D)
rw [abs_of_nonneg hb.1]
exact hb.2.trans (pow_le_one₀ (tanh_nonneg_of_beta_nonneg hbeta)
(Real.tanh_lt_one beta).le)
· simp [observedRow, hw]
lemma infiniteGibbsResponse_eq_gaussianPosteriorResponse
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (t : ℝ≥0) (v w : TreeVertex d) :
(fun p => infiniteGibbsResponse beta v w (observation t p)) =ᵐ[
μ.prod (gaussianNoiseLaw (TreeVertex d) t)] gaussianPosteriorResponse μ t v w := by
filter_upwards [hμ.infiniteGibbsCovariance_eq_posterior t v w,
hμ.infiniteGibbsSpinMean_eq_posterior t v] with p hc hm
unfold infiniteGibbsResponse gaussianPosteriorResponse posteriorResponse
rw [hc, hm]
rfl
theorem observedRow_tendsto_ae_gaussian
{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)
(hconn : ∀ n, ConnectedFinset (D n)) (hex : ∀ u, ∃ n, u ∈ D n)
(v : TreeVertex d) (hv : ∀ n, v ∈ D n) (w : TreeVertex d) (t : ℝ≥0) :
∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t),
Tendsto (fun n => observedRow beta (D n) v t w p) atTop
(𝓝 (infiniteGibbsResponse beta v w (observation t p))) := by
filter_upwards [hμ.observedRow_tendsto_ae hbeta D hD hconn v hv hex t w,
infiniteGibbsResponse_eq_gaussianPosteriorResponse hμ t v w] with p hlim hinf
rw [hinf]
exact hlim
theorem observedRow_tendsto_ae_brownian
{d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(ν : TreeVertex d → Measure Ω)
{B : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hB : ∀ i, IsPreBrownianReal (B i) (ν i))
(hmB : ∀ i t, Measurable (B i t))
(D : ℕ → Finset (TreeVertex d)) (hD : Monotone D)
(hconn : ∀ n, ConnectedFinset (D n)) (hex : ∀ u, ∃ n, u ∈ D n)
(v : TreeVertex d) (hv : ∀ n, v ∈ D n) (w : TreeVertex d) (t : ℝ≥0) :
∀ᵐ p ∂μ.prod (Measure.infinitePi ν),
Tendsto (fun n => observedRow beta (D n) v t w
(p.1, fun i => B i t (p.2 i))) atTop
(𝓝 (infiniteGibbsResponse beta v w
(fun i => (t : ℝ) * spinSign (p.1 i) + B i t (p.2 i)))) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (vertex : TreeVertex d) : IsProbabilityMeasure (ν vertex) :=
(hB vertex).isGaussianProcess.isProbabilityMeasure
let T : ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) →
(TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) :=
Prod.map id (fun omega i => B i t (omega i))
have hT : MeasurePreserving T (μ.prod (Measure.infinitePi ν))
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) :=
(MeasurePreserving.id μ).prod (brownian_array_eval_measurePreserving ν hB hmB t)
have h := observedRow_tendsto_ae_gaussian hμ hbeta D hD hconn hex v hv w t
rw [← hT.map_eq] at h
exact ae_of_ae_map hT.measurable.aemeasurable h
theorem observedRow_time_meanSquare_brownian
{d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(ν : TreeVertex d → Measure Ω)
{B : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hB : ∀ i, IsPreBrownianReal (B i) (ν i))
(hmB : ∀ i t, Measurable (B i t))
(hjoint : ∀ i, Measurable (fun z : ℝ≥0 × Ω => B i z.1 z.2))
(D : ℕ → Finset (TreeVertex d)) (hD : Monotone D)
(hconn : ∀ n, ConnectedFinset (D n)) (hex : ∀ u, ∃ n, u ∈ D n)
(v : TreeVertex d) (hv : ∀ n, v ∈ D n) (w : TreeVertex d)
(η : Measure ℝ≥0) [IsFiniteMeasure η] :
Tendsto (fun n => ∫ q : ℝ≥0 × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)),
(observedRow beta (D n) v q.1 w
(q.2.1, fun i => B i q.1 (q.2.2 i)) -
infiniteGibbsResponse beta v w
(fun i => (q.1 : ℝ) * spinSign (q.2.1 i) + B i q.1 (q.2.2 i))) ^ 2
∂η.prod (μ.prod (Measure.infinitePi ν))) atTop (𝓝 0) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (vertex : TreeVertex d) : IsProbabilityMeasure (ν vertex) :=
(hB vertex).isGaussianProcess.isProbabilityMeasure
have hs : Measurable spinSign := by intro s hs; trivial
let Y : ℝ≥0 × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) →
TreeVertex d → ℝ :=
fun q i => (q.1 : ℝ) * spinSign (q.2.1 i) + B i q.1 (q.2.2 i)
have hY : Measurable Y := by
apply Measurable.of_eval
intro i
exact ((measurable_subtype_coe.comp measurable_fst).mul
(hs.comp ((measurable_pi_apply i).comp (measurable_fst.comp measurable_snd)))).add
((hjoint i).comp (measurable_fst.prodMk
((measurable_pi_apply i).comp (measurable_snd.comp measurable_snd))))
have hX (n : ℕ) : Measurable
(fun q : ℝ≥0 × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) =>
observedRow beta (D n) v q.1 w (q.2.1, fun i => B i q.1 (q.2.2 i))) := by
classical
by_cases hw : w ∈ D n
· simp only [observedRow, ite_eq_left hw]
change Measurable ((response beta (D n) v w) ∘ Y)
exact (measurable_response beta (D n) v w).comp hY
· simpa only [observedRow, ite_eq_right hw] using (measurable_const :
Measurable (fun _ : ℝ≥0 × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) =>
(0 : ℝ)))
apply tendsto_integral_prod_sq_sub_of_bounded_ae η (μ.prod (Measure.infinitePi ν))
hX ((measurable_infiniteGibbsResponse beta v w).comp hY) (by norm_num : (0 : ℝ) ≤ 1)
· intro n
apply Eventually.of_forall
intro t
apply Eventually.of_forall
intro p
exact observedRow_abs_le_one hbeta (D n) (hconn n) v (hv n) t w _
· apply Eventually.of_forall
intro t
exact hμ.infiniteGibbsResponse_abs_le_one_brownian ν hB hmB t v w
· apply Eventually.of_forall
intro t
exact observedRow_tendsto_ae_brownian hμ hbeta ν hB hmB D hD hconn hex v hv w t
end Problem367.NormalizedResponseTime
end
end OAI
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