File size: 13,662 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 | import OAI.Probability.FreeIsing.Energy.NormalizedInfiniteTimeEnergy
import OAI.Probability.FreeIsing.Gaussian.TreeResponseVarianceMixture
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped NNReal ENNReal
namespace Problem367.NormalizedBrownianVarianceComparison
open GaussianTreeResponse
local instance : Nonempty IsingSpin := ⟨.neg⟩
variable {d : ℕ} {Ω : Type*} [MeasurableSpace Ω]
lemma shiftedRow_eq_normalized (beta : ℝ) (D : Finset (TreeVertex d))
(v w : D) (t : ℝ) (p : (TreeVertex d → IsingSpin) × (D → ℝ)) :
shiftedRow beta D v (spinDrift t) (fun _ => 0) w p =
FinitePosterior.normalizedFieldCovariance (finiteIsingEnergy beta D)
(fun eta u => spinSign (eta u)) (fun eta => spinSign (eta v))
(fun u => t * spinSign (p.1 u) + p.2 u) w := by
classical
simp only [shiftedRow, GaussianTreeResponse.finiteRow, ite_eq_left w.property]
unfold response TreePathResponse.response TreePathResponse.covariance
LeafGibbs.covariance TreePathResponse.mean
FinitePosterior.normalizedFieldCovariance FinitePosterior.fieldCovariance
simp only [extendFiniteSpin_apply D _ v, extendFiniteSpin_apply D _ w,
GibbsLeaf.expectation_eq_fieldMean]
have hx : D.restrict (fun u => spinDrift t p.1 u +
Function.updateFinset (fun _ => (0 : ℝ)) D p.2 u) =
(fun u : D => t * spinSign (p.1 u) + p.2 u) := by
funext u
simp [Finset.restrict, spinDrift, Function.updateFinset]
rw [hx]
lemma shiftedRaw_eq_covariance (beta : ℝ) (D : Finset (TreeVertex d))
(v w : D) (t : ℝ) (p : (TreeVertex d → IsingSpin) × (D → ℝ)) :
GaussianTreeCovariance.finiteRow beta D v (spinDrift t p.1) (fun _ => 0) w p.2 =
FinitePosterior.fieldCovariance (finiteIsingEnergy beta D)
(fun eta u => spinSign (eta u)) (fun eta => spinSign (eta v))
(fun u => t * spinSign (p.1 u) + p.2 u) w := by
classical
simp only [GaussianTreeCovariance.finiteRow, ite_eq_left w.property]
unfold GaussianTreeCovariance.covariance LeafGibbs.covariance
FinitePosterior.fieldCovariance
simp only [extendFiniteSpin_apply D _ v, extendFiniteSpin_apply D _ w,
GibbsLeaf.expectation_eq_fieldMean]
have hx : D.restrict (fun u => spinDrift t p.1 u +
Function.updateFinset (fun _ => (0 : ℝ)) D p.2 u) =
(fun u : D => t * spinSign (p.1 u) + p.2 u) := by
funext u
simp [Finset.restrict, spinDrift, Function.updateFinset]
rw [hx]
theorem varianceLower_mul_integral_normalized_sq_le_raw
(μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ]
(ν : TreeVertex d → Measure Ω) {beta : ℝ} (hb : 0 ≤ beta)
{B : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hB : ∀ i, IsPreBrownianReal (B i) (ν i))
(hmB : ∀ i t, Measurable (B i t))
(D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v w : D)
(hv : v.1.1 = []) {T : ℝ} (hT : 0 ≤ T) {t : ℝ≥0} (ht : (t : ℝ) ≤ T) :
GaussianVarianceLower.varianceLower T (T + (d : ℝ) * beta) *
(∫ p, finiteIsingBrownianNormalizedCovariance beta D v w B t p ^ 2
∂μ.prod (Measure.infinitePi ν)) ≤
∫ p, finiteIsingBrownianCovariance beta D v w B t p ^ 2
∂μ.prod (Measure.infinitePi ν) := by
have (i : TreeVertex d) : IsProbabilityMeasure (ν i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
have hbound (σ : TreeVertex d → IsingSpin) : |spinDrift (t : ℝ) σ v| ≤ T := by
unfold spinDrift
cases σ v <;> simpa [spinSign, abs_of_nonneg t.property] using ht
have hc := varianceLower_mul_integral_shiftedRow_sq_le_raw μ hb (fun _ => t)
hT hD v hv ht (spinDrift t) (measurable_spinDrift d t) hT hbound (fun _ => 0) w
have hp := (MeasurePreserving.id μ).prod
(finiteBrownianNoise_restrict_measurePreserving ν hB hmB D t)
let f : ((TreeVertex d → IsingSpin) × (D → ℝ)) → ℝ := fun p =>
FinitePosterior.normalizedFieldCovariance (finiteIsingEnergy beta D)
(fun eta u => spinSign (eta u)) (fun eta => spinSign (eta v))
(fun u => (t : ℝ) * spinSign (p.1 u) + p.2 u) w ^ 2
let g : ((TreeVertex d → IsingSpin) × (D → ℝ)) → ℝ := fun p =>
FinitePosterior.fieldCovariance (finiteIsingEnergy beta D)
(fun eta u => spinSign (eta u)) (fun eta => spinSign (eta v))
(fun u => (t : ℝ) * spinSign (p.1 u) + p.2 u) w ^ 2
have hobs : Measurable (fun p : (TreeVertex d → IsingSpin) × (D → ℝ) =>
fun u : D => (t : ℝ) * spinSign (p.1 u) + p.2 u) := by
fun_prop
have hbinary (eta : D → IsingSpin) : spinSign (eta v) = -1 ∨ spinSign (eta v) = 1 := by
cases eta v <;> simp [spinSign]
have hf : Measurable f :=
(((FinitePosterior.continuous_normalizedFieldCovariance_entropy _ _ _ w hbinary
⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩).measurable.comp hobs).pow_const 2)
have hg : Measurable g :=
(((FinitePosterior.continuous_fieldCovariance _ _ _ w).measurable.comp hobs).pow_const 2)
simp_rw [shiftedRow_eq_normalized, shiftedRaw_eq_covariance] at hc
have heF := hp.hasLaw.integral_comp hf.aestronglyMeasurable
have heG := hp.hasLaw.integral_comp hg.aestronglyMeasurable
change (∫ p, f (Prod.map id (fun ω (u : D) => B u t (ω u)) p)
∂μ.prod (Measure.infinitePi ν)) = _ at heF
change (∫ p, g (Prod.map id (fun ω (u : D) => B u t (ω u)) p)
∂μ.prod (Measure.infinitePi ν)) = _ at heG
have hobsEq (p : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) :
finiteIsingBrownianObservation B D (t : ℝ) p =
(fun u : D => (t : ℝ) * spinSign (p.1 u) + B u t (p.2 u)) := by
funext u
simp only [finiteIsingBrownianObservation, Real.toNNReal_coe]
simp only [finiteIsingBrownianNormalizedCovariance, finiteIsingBrownianCovariance]
simp_rw [hobsEq]
simpa only [f, g, Prod.map_fst, Prod.map_snd, id_eq]
using (show GaussianVarianceLower.varianceLower T (T + (d : ℝ) * beta) *
(∫ p, f (Prod.map id (fun ω (u : D) => B u t (ω u)) p)
∂μ.prod (Measure.infinitePi ν)) ≤
∫ p, g (Prod.map id (fun ω (u : D) => B u t (ω u)) p)
∂μ.prod (Measure.infinitePi ν) by rw [heF, heG]; exact hc)
lemma integrable_coefficients_sq
(μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ]
(ν : TreeVertex d → Measure Ω) [∀ i, IsProbabilityMeasure (ν i)]
{B : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hmB : ∀ i t, Measurable (B i t)) (beta : ℝ)
(D : Finset (TreeVertex d)) (v w : D) (t : ℝ) :
Integrable (fun p => finiteIsingBrownianNormalizedCovariance beta D v w B t p ^ 2)
(μ.prod (Measure.infinitePi ν)) ∧
Integrable (fun p => finiteIsingBrownianCovariance beta D v w B t p ^ 2)
(μ.prod (Measure.infinitePi ν)) := by
have hY : Measurable (finiteIsingBrownianObservation B D t) := by
apply Measurable.of_eval
intro u
exact (((measurable_of_finite spinSign).comp
((measurable_pi_apply (u : TreeVertex d)).comp measurable_fst)).const_mul t).add
((hmB u t.toNNReal).comp ((measurable_pi_apply (u : TreeVertex d)).comp measurable_snd))
have hs (eta : D → IsingSpin) (u : D) : spinSign (eta u) = -1 ∨ spinSign (eta u) = 1 := by
cases eta u <;> simp [spinSign]
have hb (eta : D → IsingSpin) (u : D) : |spinSign (eta u)| ≤ 1 := by
cases eta u <;> norm_num [spinSign]
constructor
· apply Integrable.of_bound
(((FinitePosterior.continuous_normalizedFieldCovariance_entropy (finiteIsingEnergy beta D)
(fun eta u => spinSign (eta u)) (fun eta => spinSign (eta v)) w
(fun eta => hs eta v) ⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩).measurable.comp
hY).pow_const 2).aestronglyMeasurable 1
exact Filter.Eventually.of_forall fun p => by
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _), sq_le_one_iff_abs_le_one]
exact FinitePosterior.abs_normalizedFieldCovariance_le_one_entropy _ _ _ _ w
(fun eta => hs eta v) ⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩ (fun eta => hb eta w)
· apply Integrable.of_bound
(((FinitePosterior.continuous_fieldCovariance (finiteIsingEnergy beta D)
(fun eta u => spinSign (eta u)) (fun eta => spinSign (eta v)) w).measurable.comp
hY).pow_const 2).aestronglyMeasurable 4
exact Filter.Eventually.of_forall fun p => by
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
have h := FinitePosterior.abs_fieldCovariance_le (finiteIsingEnergy beta D)
(fun eta u => spinSign (eta u)) (fun eta => spinSign (eta v))
(finiteIsingBrownianObservation B D t p) w 1 (fun eta => hs eta w) (fun eta => hb eta v)
change finiteIsingBrownianCovariance beta D v w B t p ^ 2 ≤ 4
have hsq := (sq_le_sq₀ (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 2)).mpr
(show |finiteIsingBrownianCovariance beta D v w B t p| ≤ 2 by simpa [finiteIsingBrownianCovariance] using h)
norm_num only [sq_abs, Nat.reducePow, OfNat.ofNat] at hsq
nlinarith [hsq]
theorem varianceLower_mul_integral_energy_le_information
{beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 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 : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v : D)
(hv : v.1.1 = []) {H T : ℝ} (hH : 0 ≤ H) (hT : 0 ≤ T) (hTH : T ≤ H) :
GaussianVarianceLower.varianceLower H (H + (d : ℝ) * beta) *
(∫ t in 0..T, ∫ p, ∑ w : D,
finiteIsingBrownianNormalizedCovariance beta D v w B t p ^ 2
∂μ.prod (Measure.infinitePi ν)) ≤
∫ p, finiteIsingBrownianMean beta D v B T p ^ 2 ∂μ.prod (Measure.infinitePi ν) := by
classical
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (i : TreeVertex d) : IsProbabilityMeasure (ν i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
have hn := hμ.integral_finiteIsingBrownianNormalizedEnergy_le hb ν hB hmB D hD v hT
have hr := hμ.integral_finiteIsingBrownianEnergy ν hB hmB D hD v hT
have hpoint (t : ℝ) (ht : t ∈ Icc 0 T) :
GaussianVarianceLower.varianceLower H (H + (d : ℝ) * beta) *
(∫ p, ∑ w : D, finiteIsingBrownianNormalizedCovariance beta D v w B t p ^ 2
∂μ.prod (Measure.infinitePi ν)) ≤
∫ p, ∑ w : D, finiteIsingBrownianCovariance beta D v w B t p ^ 2
∂μ.prod (Measure.infinitePi ν) := by
rw [integral_finsetSum _ (fun w _ => (integrable_coefficients_sq μ ν hmB beta D v w t).1),
integral_finsetSum _ (fun w _ => (integrable_coefficients_sq μ ν hmB beta D v w t).2),
Finset.mul_sum]
apply Finset.sum_le_sum
intro w _
have hh := varianceLower_mul_integral_normalized_sq_le_raw μ ν hb hB hmB
D hD v w hv hH (t := t.toNNReal) (by simpa only [Real.coe_toNNReal t ht.1] using ht.2.trans hTH)
simpa only [Real.coe_toNNReal t ht.1] using hh
have hi := intervalIntegral.integral_mono_on hT
(hn.1.const_mul (GaussianVarianceLower.varianceLower H (H + (d : ℝ) * beta))) hr.1 hpoint
rw [intervalIntegral.integral_const_mul, hr.2.1] at hi
exact hi
theorem lintegral_energy_le_information
{beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 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 : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v : D)
(hv : v.1.1 = []) {H T : ℝ} (hH : 0 ≤ H) (hT : 0 ≤ T) (hTH : T ≤ H) :
(∫⁻ t in Ioc 0 T, ∫⁻ p,
ENNReal.ofReal (∑ w : D, finiteIsingBrownianNormalizedCovariance beta D v w B t p ^ 2)
∂μ.prod (Measure.infinitePi ν)) ≤
ENNReal.ofReal ((∫ p, finiteIsingBrownianMean beta D v B T p ^ 2
∂μ.prod (Measure.infinitePi ν)) /
GaussianVarianceLower.varianceLower H (H + (d : ℝ) * beta)) := by
classical
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (i : TreeVertex d) : IsProbabilityMeasure (ν i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
let ρ := μ.prod (Measure.infinitePi ν)
let R := fun t p => ∑ w : D, finiteIsingBrownianNormalizedCovariance beta D v w B t p ^ 2
let E := fun t => ∫ p, R t p ∂ρ
have hRpos (t : ℝ) (p) : 0 ≤ R t p := Finset.sum_nonneg fun w _ => sq_nonneg _
have hRI (t : ℝ) : Integrable (R t) ρ :=
integrable_finsetSum _ (fun w _ => (integrable_coefficients_sq μ ν hmB beta D v w t).1)
have hinner (t : ℝ) : (∫⁻ p, ENNReal.ofReal (R t p) ∂ρ) = ENNReal.ofReal (E t) :=
(ofReal_integral_eq_lintegral_ofReal (hRI t) (Filter.Eventually.of_forall (hRpos t))).symm
have hn := hμ.integral_finiteIsingBrownianNormalizedEnergy_le hb ν hB hmB D hD v hT
have hEI : Integrable E (volume.restrict (Ioc 0 T)) :=
(intervalIntegrable_iff_integrableOn_Ioc_of_le hT).mp hn.1
change (∫⁻ t in Ioc 0 T, ∫⁻ p, ENNReal.ofReal (R t p) ∂ρ) ≤ _
simp_rw [hinner]
rw [← ofReal_integral_eq_lintegral_ofReal hEI
(Filter.Eventually.of_forall fun t => integral_nonneg (hRpos t)),
← intervalIntegral.integral_of_le hT]
apply ENNReal.ofReal_le_ofReal
apply (le_div_iff₀ (GaussianVarianceLower.varianceLower_pos _ _)).2
simpa only [mul_comm] using
varianceLower_mul_integral_energy_le_information hμ hb ν hB hmB D hD v hv hH hT hTH
end Problem367.NormalizedBrownianVarianceComparison
end
end OAI
|