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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 | import OAI.Probability.FreeIsing.Ising.CanonicalContinuousFields
import OAI.Probability.FreeIsing.Ising.CopyResponseRowPaths
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology BigOperators
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation
variable {d : ℕ} {beta : ℝ}
private lemma continuous_tanh : Continuous Real.tanh :=
continuous_iff_continuousAt.mpr (fun z => (hasDerivAt_tanh_response z).continuousAt)
def siteRootResponse (hb : 0 ≤ beta) (v : TreeVertex d)
(H : TreeVertex d → ℝ) : ℝ :=
Real.tanh beta ^ v.1.length *
∏ i ∈ Finset.range v.1.length, responseAttenuation (Real.tanh beta)
(CavityArray.field (tanh_nonneg_of_beta_nonneg hb) (Real.tanh_lt_one beta)
H (ConnectedHull.takeVertex v (i + 1)) (ConnectedHull.takeVertex v i))
lemma continuous_siteRootResponse (hb : 0 ≤ beta) (v : TreeVertex d) :
Continuous (siteRootResponse hb v) := by
unfold siteRootResponse
apply Continuous.const_mul
apply continuous_finsetProd
intro i hi
exact (continuous_responseAttenuation (Real.tanh_sq_lt_one beta).le).comp
(continuous_fst.comp ((continuous_cavityFromSites
(tanh_nonneg_of_beta_nonneg hb) (Real.tanh_lt_one beta)).comp
((continuous_apply _).prodMk (continuous_apply _))))
lemma siteRootResponse_mem_Icc (hb : 0 ≤ beta) (v : TreeVertex d)
(H : TreeVertex d → ℝ) :
0 ≤ siteRootResponse hb v H ∧ siteRootResponse hb v H ≤ 1 := by
have hp0 : 0 ≤ ∏ i ∈ Finset.range v.1.length, responseAttenuation (Real.tanh beta)
(CavityArray.field (tanh_nonneg_of_beta_nonneg hb) (Real.tanh_lt_one beta)
H (ConnectedHull.takeVertex v (i + 1)) (ConnectedHull.takeVertex v i)) :=
Finset.prod_nonneg fun i _ => (responseAttenuation_pos (Real.tanh_sq_lt_one beta).le _).le
have hp1 : (∏ i ∈ Finset.range v.1.length, responseAttenuation (Real.tanh beta)
(CavityArray.field (tanh_nonneg_of_beta_nonneg hb) (Real.tanh_lt_one beta)
H (ConnectedHull.takeVertex v (i + 1)) (ConnectedHull.takeVertex v i))) ≤ 1 :=
Finset.prod_le_one₀
(fun i _ => (responseAttenuation_pos (Real.tanh_sq_lt_one beta).le _).le)
(fun i _ => responseAttenuation_le_one (Real.tanh_sq_lt_one beta).le _)
have ht0 := pow_nonneg (tanh_nonneg_of_beta_nonneg hb) v.1.length
have ht1 : Real.tanh beta ^ v.1.length ≤ 1 :=
pow_le_one₀ (tanh_nonneg_of_beta_nonneg hb) (Real.tanh_lt_one beta).le
exact ⟨mul_nonneg ht0 hp0, (mul_le_mul ht1 hp1 hp0 (by norm_num)).trans_eq (one_mul 1)⟩
def continuousRootResponse (hb : 0 ≤ beta) (v : TreeVertex d)
(t : ℝ≥0) (x : Paths d) : ℝ :=
siteRootResponse hb v (fun i => continuousField beta i t x)
lemma continuousRootResponse_progressive (hb : 0 ≤ beta) (v : TreeVertex d) :
IsStronglyProgressive (filtration d) (continuousRootResponse hb v) := by
intro t
let : MeasurableSpace (Paths d) := filtration d t
exact ((continuous_siteRootResponse hb v).measurable.comp
(Measurable.of_eval (fun i => (continuousField_progressive beta i t).measurable))).stronglyMeasurable
lemma continuousRootResponse_abs_le_one (hb : 0 ≤ beta) (v : TreeVertex d)
(t : ℝ≥0) (x : Paths d) : |continuousRootResponse hb v t x| ≤ 1 := by
obtain ⟨h0, h1⟩ := siteRootResponse_mem_Icc hb v (fun i => continuousField beta i t x)
exact (abs_of_nonneg h0).trans_le h1
theorem continuousRootResponse_continuous_ae
(hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (v : TreeVertex d) :
∀ᵐ x ∂observationLaw μ, Continuous (fun t => continuousRootResponse hb v t x) := by
have hc (i : TreeVertex d) := (continuousField_modification hd hμ hb hcrit i).1
filter_upwards [ae_all_iff.mpr hc] with x hx
exact (continuous_siteRootResponse hb v).comp (continuous_pi hx)
theorem continuousRootResponse_eq_response_ae
(hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(v : TreeVertex d) (t : ℝ≥0) :
continuousRootResponse hb v t =ᵐ[observationLaw μ]
fun x => infiniteGibbsResponse beta (ConnectedHull.root d) v (fun i => x i t) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have he (i : TreeVertex d) := (continuousField_modification hd hμ hb hcrit i).2 t
filter_upwards [ae_all_iff.mpr he,
IsingCopyResponseRowPaths.row_eq_product_ae hμ hb t v] with x hx hr
change siteRootResponse hb v (fun i => continuousField beta i t x) = _
rw [show (fun i => continuousField beta i t x) = (fun i => field beta i t x)
from funext hx]
exact hr.symm
def continuousRootCovariance (hb : 0 ≤ beta) (v : TreeVertex d)
(t : ℝ≥0) (x : Paths d) : ℝ :=
(1 - Real.tanh (continuousField beta (ConnectedHull.root d) t x) ^ 2) *
continuousRootResponse hb v t x
lemma continuousRootCovariance_progressive (hb : 0 ≤ beta) (v : TreeVertex d) :
IsStronglyProgressive (filtration d) (continuousRootCovariance hb v) := by
intro t
have hf := continuousField_progressive beta (ConnectedHull.root d) t
exact (stronglyMeasurable_const.sub
((continuous_tanh.comp_stronglyMeasurable hf).pow 2)).mul
(continuousRootResponse_progressive hb v t)
lemma continuousRootCovariance_abs_le_one (hb : 0 ≤ beta) (v : TreeVertex d)
(t : ℝ≥0) (x : Paths d) : |continuousRootCovariance hb v t x| ≤ 1 := by
have h0 := (sub_pos.mpr (Real.tanh_sq_lt_one
(continuousField beta (ConnectedHull.root d) t x))).le
have h1 : 1 - Real.tanh (continuousField beta (ConnectedHull.root d) t x) ^ 2 ≤ 1 :=
sub_le_self _ (sq_nonneg _)
rw [continuousRootCovariance, abs_mul, abs_of_nonneg h0]
exact (mul_le_mul h1 (continuousRootResponse_abs_le_one hb v t x)
(abs_nonneg _) (by norm_num)).trans_eq (one_mul 1)
theorem continuousRootCovariance_continuous_ae
(hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (v : TreeVertex d) :
∀ᵐ x ∂observationLaw μ, Continuous (fun t => continuousRootCovariance hb v t x) := by
filter_upwards [(continuousField_modification hd hμ hb hcrit (ConnectedHull.root d)).1,
continuousRootResponse_continuous_ae hd hμ hb hcrit v] with x hf hr
exact (continuous_const.sub ((continuous_tanh.comp hf).pow 2)).mul hr
theorem continuousRootCovariance_eq_covariance_ae
(hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(v : TreeVertex d) (t : ℝ≥0) :
continuousRootCovariance hb v t =ᵐ[observationLaw μ]
fun x => infiniteGibbsCovariance beta (ConnectedHull.root d) v (fun i => x i t) := by
filter_upwards [(continuousField_modification hd hμ hb hcrit (ConnectedHull.root d)).2 t,
continuousRootResponse_eq_response_ae hd hμ hb hcrit v t,
tanh_field_eq_posterior_ae hμ hb (ConnectedHull.root d) t,
posterior_mem_Ioo_ae hμ hb (ConnectedHull.root d) t] with x hf hr hm hi
have hpos : 0 < 1 - posterior beta (ConnectedHull.root d) t x ^ 2 := by
nlinarith [hi.1, hi.2]
rw [continuousRootCovariance, hf, hm, hr, infiniteGibbsResponse]
change (1 - posterior beta (ConnectedHull.root d) t x ^ 2) *
(_ / (1 - posterior beta (ConnectedHull.root d) t x ^ 2)) = _
exact mul_div_cancel₀ _ hpos.ne'
end Problem367.IsingObservationFactor
end
end OAI
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