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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 | import OAI.Probability.FreeIsing.Gaussian.TreeResponsePlanted
import OAI.Probability.FreeIsing.Gaussian.TreeResponsePosterior
import OAI.Probability.FreeIsing.Posterior.NormalizedPosteriorLimit
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped Topology NNReal
namespace Problem367
open PlantedSymmetry GaussianTreeResponse GaussianTreeCovariance
lemma GaussianTreeResponse.measurable_observedRow {d : ℕ}
(beta : ℝ) (D : Finset (TreeVertex d)) (r : TreeVertex d)
(t : ℝ) (v : TreeVertex d) : Measurable (observedRow beta D r t v) := by
classical
unfold observedRow
by_cases hv : v ∈ D
· simp only [ite_eq_left hv]
exact (measurable_response beta D r v).comp (measurable_observation t)
· simp only [ite_eq_right hv]
exact measurable_const
theorem HasFreeIsingLaw.observedRow_eq_posteriorResponse
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ)
(D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (r v : TreeVertex d)
(hr : r ∈ D) (hv : v ∈ D) (t : ℝ≥0) :
observedRow beta D r t v =ᵐ[μ.prod (gaussianNoiseLaw (TreeVertex d) t)]
posteriorResponse (μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(MeasurableSpace.comap (finiteObservation D t) inferInstance)
(fun p => spinSign (p.1 r)) (fun p => spinSign (p.1 v)) := by
have h := hμ.finiteObservation_normalizedResponse D hD t ⟨r, hr⟩ ⟨v, hv⟩
filter_upwards [h] with p hp
simpa only [observedRow, ite_eq_left hv, posteriorResponse, posteriorCovariance] using hp.symm
theorem HasFreeIsingLaw.observedRow_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)
(hconn : ∀ n, ConnectedFinset (D n)) (r : TreeVertex d)
(hr : ∀ n, r ∈ D n) (hex : ∀ v, ∃ n, v ∈ D n) (t : ℝ≥0) (v : TreeVertex d) :
∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t),
Tendsto (fun n => observedRow beta (D n) r t v p) atTop
(𝓝 (gaussianPosteriorResponse μ t r v p)) := by
have heq (n : ℕ) : ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t),
v ∈ D n → observedRow beta (D n) r t v p =
posteriorResponse (μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(MeasurableSpace.comap (finiteObservation (D n) t) inferInstance)
(fun p => spinSign (p.1 r)) (fun p => spinSign (p.1 v)) p := by
by_cases hv : v ∈ D n
· exact (hμ.observedRow_eq_posteriorResponse (D n) (hconn n) r v (hr n) hv t).mono
(fun p hp _ => hp)
· exact Filter.Eventually.of_forall (fun _ hv' => (hv hv').elim)
obtain ⟨N, hN⟩ := hex v
have hmem : ∀ᶠ n in atTop, v ∈ D n :=
eventually_atTop.2 ⟨N, fun n hn => hD hn hN⟩
filter_upwards [ae_all_iff.mpr heq, hμ.gaussian_posteriorResponse_tendsto_ae hbeta D hD hex t r v] with p hp hlim
exact hlim.congr' (hmem.mono (fun n hn => (hp n hn).symm))
theorem HasFreeIsingLaw.observedRow_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)
(hconn : ∀ n, ConnectedFinset (D n)) (r : TreeVertex d)
(hr : ∀ n, r ∈ D n) (hex : ∀ v, ∃ n, v ∈ D n) (t : ℝ≥0) (v : TreeVertex d) :
Tendsto (fun n => ∫ p,
(observedRow beta (D n) r t v p - gaussianPosteriorResponse μ t r v p) ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) atTop (𝓝 0) := by
apply (hμ.gaussian_posteriorResponse_meanSquare hbeta D hD hex t r v).congr'
obtain ⟨N, hN⟩ := hex v
filter_upwards [eventually_ge_atTop N] with n hn
apply integral_congr_ae
filter_upwards [hμ.observedRow_eq_posteriorResponse (D n) (hconn n) r v (hr n) (hD hn hN) t] with p hp
rw [hp]
theorem HasFreeIsingLaw.observedRow_full_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)
(hconn : ∀ n, ConnectedFinset (D n)) (r : TreeVertex d)
(hr : ∀ n, r ∈ D n) (hrnil : r.1 = [])
(hex : ∀ v, ∃ n, v ∈ D n) {t : ℝ≥0} (ht : 0 < t)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) :
(∀ n, Summable (fun v => ∫ p,
(observedRow beta (D n) r t v p - gaussianPosteriorResponse μ t r v p) ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t))) ∧
Tendsto (fun n => ∑' v, ∫ p,
(observedRow beta (D n) r t v p - gaussianPosteriorResponse μ t r v p) ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) atTop (𝓝 0) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
let P := μ.prod (gaussianNoiseLaw (TreeVertex d) t)
let q := Real.tanh beta ^ 2 * gaussianContraction beta t
let X := fun n v => observedRow beta (D n) r t v
let Y := fun v => gaussianPosteriorResponse μ t r v
have hq : 0 ≤ q := mul_nonneg (sq_nonneg _) (gaussianContraction_pos beta t).le
have hqs : ((d - 1 : ℕ) : ℝ) * q < 1 := gaussian_radial_ratio_lt_one ht hcrit
have hXm (n : ℕ) (v : TreeVertex d) : AEStronglyMeasurable (X n v) P :=
(measurable_observedRow beta (D n) r t v).aestronglyMeasurable
have hYm (v : TreeVertex d) : AEStronglyMeasurable (Y v) P :=
((stronglyMeasurable_posteriorResponse P (observationSigma (TreeVertex d) t)
(fun p => spinSign (p.1 r)) (fun p => spinSign (p.1 v))).mono
(observationSigma_le (TreeVertex d) t)).aestronglyMeasurable
have hXi (n : ℕ) (v : TreeVertex d) : Integrable (fun p => X n v p ^ 2) P :=
integrable_observedRow_sq μ hbeta t (hconn n) (hr n) hrnil v
have hXb (n : ℕ) (v : TreeVertex d) : (∫ p, X n v p ^ 2 ∂P) ≤ q ^ v.1.length :=
integral_observedRow_sq_le μ hbeta ht (hconn n) (hr n) hrnil v
have hY (v : TreeVertex d) : Integrable (fun p => Y v p ^ 2) P ∧
(∫ p, Y v p ^ 2 ∂P) ≤ q ^ v.1.length :=
integrable_sq_and_integral_le_of_tendsto_ae P
(fun n => hXm n v) (hYm v) (pow_nonneg hq _)
(hμ.observedRow_tendsto_ae hbeta D hD hconn r hr hex t v)
(Filter.Eventually.of_forall (fun n => ⟨hXi n v, hXb n v⟩))
have hEi (n : ℕ) (v : TreeVertex d) :
Integrable (fun p => (X n v p - Y v p) ^ 2) P := by
exact (((memLp_two_iff_integrable_sq (hXm n v)).mpr (hXi n v)).sub
((memLp_two_iff_integrable_sq (hYm v)).mpr (hY v).1)).integrable_sq
have hEb (n : ℕ) (v : TreeVertex d) :
(∫ p, (X n v p - Y v p) ^ 2 ∂P) ≤ 4 * q ^ v.1.length := by
have hle : (∫ p, (X n v p - Y v p) ^ 2 ∂P) ≤
∫ p, 2 * X n v p ^ 2 + 2 * Y v p ^ 2 ∂P := by
apply integral_mono (hEi n v) (((hXi n v).const_mul 2).add ((hY v).1.const_mul 2))
intro p
change (X n v p - Y v p) ^ 2 ≤ 2 * X n v p ^ 2 + 2 * Y v p ^ 2
nlinarith [sq_nonneg (X n v p + Y v p)]
rw [integral_add ((hXi n v).const_mul 2) ((hY v).1.const_mul 2),
integral_const_mul, integral_const_mul] at hle
linarith [hXb n v, (hY v).2]
have hs : Summable (fun v : TreeVertex d => 4 * q ^ v.1.length) :=
(summable_geometric_radial d hq hqs).mul_left 4
refine ⟨fun n => Summable.of_nonneg_of_le
(fun v => integral_nonneg (fun p => sq_nonneg _)) (hEb n) hs, ?_⟩
have hlim := tendsto_tsum_of_dominated_convergence hs
(fun v => hμ.observedRow_meanSquare hbeta D hD hconn r hr hex t v)
(Filter.Eventually.of_forall (fun n v => by
rw [Real.norm_of_nonneg (integral_nonneg (fun p => sq_nonneg _))]
exact hEb n v))
simpa only [tsum_zero] using hlim
end Problem367
end
end OAI
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