File size: 8,210 Bytes
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.FiniteIsingGaussianEnergy
import OAI.Probability.FreeIsing.Stochastic.FiniteFieldEntropyIntegral
import OAI.Probability.FreeIsing.Trees.GibbsField
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators NNReal
namespace Problem367
local instance finiteIsingFieldEnergyNonemptySpin : Nonempty IsingSpin := ⟨.neg⟩
theorem finiteIsingEntropy_le_own_observation {d : ℕ} {beta : ℝ}
(hbeta : 0 ≤ beta) (D : Finset (TreeVertex d)) (v : D) (y : D → ℝ) :
FinitePosterior.fieldEntropy (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) w => spinSign (eta w))
(fun eta => spinSign (eta v)) y ≤ 2 * (|y v| + (d : ℝ) * beta) := by
classical
have hspin : ∀ eta : D → IsingSpin, spinSign (eta v) = -1 ∨ spinSign (eta v) = 1 := by
intro eta
cases eta v <;> simp [spinSign]
have hneg : ∃ eta : D → IsingSpin, spinSign (eta v) = -1 := ⟨fun _ => .neg, rfl⟩
have hpos : ∃ eta : D → IsingSpin, spinSign (eta v) = 1 := ⟨fun _ => .pos, rfl⟩
have hF := FinitePosterior.fieldEntropy_le_abs_field (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) w => spinSign (eta w))
(fun eta => spinSign (eta v)) y hspin hneg hpos
have hb := finite_tree_gibbs_field_uniform_bound D hbeta y v
change |FinitePosterior.fieldField (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) w => spinSign (eta w))
(fun eta => spinSign (eta v)) y - y v| ≤ (d : ℝ) * beta at hb
have ht := norm_le_norm_sub_add (FinitePosterior.fieldField (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) w => spinSign (eta w))
(fun eta => spinSign (eta v)) y) (y v)
simp only [Real.norm_eq_abs] at ht
linarith
theorem HasFreeIsingLaw.integral_finiteIsingEntropy_le_own_observation
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(D : Finset (TreeVertex d)) (v : D) (t : ℝ) :
let spin : (D → IsingSpin) → D → ℝ := fun eta w => spinSign (eta w)
let ρ := (μ.map D.restrict).prod (Measure.pi (fun _ : D => gaussianReal 0 1))
(∫ x, FinitePosterior.fieldEntropy (finiteIsingEnergy beta D) spin
(fun eta => spinSign (eta v)) (FinitePosterior.finiteStandardObservation spin t x) ∂ρ) ≤
2 * ((∫ x, |FinitePosterior.finiteStandardObservation spin t x v| ∂ρ) + (d : ℝ) * beta) := by
classical
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
let spin : (D → IsingSpin) → D → ℝ := fun eta w => spinSign (eta w)
let ρ := (μ.map D.restrict).prod (Measure.pi (fun _ : D => gaussianReal 0 1))
have hcoords : ∀ eta w, spin eta w = -1 ∨ spin eta w = 1 := by
intro eta w
change spinSign (eta w) = -1 ∨ spinSign (eta w) = 1
cases eta w <;> simp [spinSign]
have hobsInt : Integrable (FinitePosterior.finiteStandardObservation spin t) ρ :=
FinitePosterior.integrable_finiteStandardObservation (μ.map D.restrict) spin hcoords t
have hcoordInt : Integrable (fun x => |FinitePosterior.finiteStandardObservation spin t x v|) ρ :=
(hobsInt.eval v).abs
have hFInt : Integrable (fun x => FinitePosterior.fieldEntropy (finiteIsingEnergy beta D) spin
(fun eta => spinSign (eta v)) (FinitePosterior.finiteStandardObservation spin t x)) ρ :=
FinitePosterior.integrable_finiteStandardEntropy (μ.map D.restrict) (finiteIsingEnergy beta D)
spin (fun eta => spinSign (eta v)) (fun eta => hcoords eta v)
⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩ hcoords t
have hb := integral_mono hFInt
((hcoordInt.add (integrable_const ((d : ℝ) * beta))).const_mul 2)
(fun x => finiteIsingEntropy_le_own_observation hbeta D v
(FinitePosterior.finiteStandardObservation spin t x))
change (∫ x, FinitePosterior.fieldEntropy (finiteIsingEnergy beta D) spin
(fun eta => spinSign (eta v)) (FinitePosterior.finiteStandardObservation spin t x) ∂ρ) ≤ _
calc
_ ≤ ∫ x, 2 * (|FinitePosterior.finiteStandardObservation spin t x v| + (d : ℝ) * beta) ∂ρ := hb
_ = _ := by rw [integral_const_mul, integral_add hcoordInt (integrable_const _)]; simp [ρ, spin]
theorem finiteIsingEntropy_zero {d : ℕ} (beta : ℝ) (D : Finset (TreeVertex d)) (v : D) :
FinitePosterior.fieldEntropy (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) w => spinSign (eta w))
(fun eta => spinSign (eta v)) 0 = 0 := by
have hz := finiteIsingPosterior_spin_zero beta D v
change FinitePosterior.fieldMean (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) w => spinSign (eta w))
(fun eta => spinSign (eta v)) 0 = 0 at hz
simp [FinitePosterior.fieldEntropy, hz]
theorem HasFreeIsingLaw.integral_finiteIsingEntropyEnergy
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ)
(D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v : D)
{T : ℝ} (hT : 0 ≤ T) :
let spin : (D → IsingSpin) → D → ℝ := fun eta w => spinSign (eta w)
let ρ := (μ.map D.restrict).prod (Measure.pi (fun _ : D => gaussianReal 0 1))
let C : ℝ → ℝ := fun t => ∫ x,
(1 + FinitePosterior.fieldMean (finiteIsingEnergy beta D) spin
(fun eta => spinSign (eta v)) (FinitePosterior.finiteStandardObservation spin t x) ^ 2) *
∑ w, FinitePosterior.normalizedFieldCovariance (finiteIsingEnergy beta D) spin
(fun eta => spinSign (eta v)) (FinitePosterior.finiteStandardObservation spin t x) w ^ 2 ∂ρ
IntervalIntegrable C volume 0 T ∧
(∫ t in 0..T, C t) =
(∫ x, FinitePosterior.fieldEntropy (finiteIsingEnergy beta D) spin
(fun eta => spinSign (eta v)) (FinitePosterior.finiteStandardObservation spin T x) ∂ρ) ∧
0 ≤ (∫ t in 0..T, C t) := by
classical
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have hcoords : ∀ (eta : D → IsingSpin) w, spinSign (eta w) = -1 ∨ spinSign (eta w) = 1 := by
intro eta w
cases eta w <;> simp [spinSign]
have h := FinitePosterior.integral_finiteStandardEntropyEnergy
(μ.map D.restrict) (finiteIsingEnergy beta D)
(hμ.restrict_real_eq_normalized_exp D hD)
(fun (eta : D → IsingSpin) w => spinSign (eta w)) (fun eta => spinSign (eta v))
(fun eta => hcoords eta v) ⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩ hcoords hT
simpa only [finiteIsingEntropy_zero, sub_zero] using h
theorem HasFreeIsingLaw.integral_finiteIsingNormalizedEnergy_le
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v : D)
{T : ℝ} (hT : 0 ≤ T) :
let spin : (D → IsingSpin) → D → ℝ := fun eta w => spinSign (eta w)
let ρ := (μ.map D.restrict).prod (Measure.pi (fun _ : D => gaussianReal 0 1))
let Q : ℝ → ℝ := fun t => ∫ x, ∑ w,
FinitePosterior.normalizedFieldCovariance (finiteIsingEnergy beta D) spin
(fun eta => spinSign (eta v)) (FinitePosterior.finiteStandardObservation spin t x) w ^ 2 ∂ρ
IntervalIntegrable Q volume 0 T ∧
(∫ t in 0..T, Q t) ≤
2 * ((∫ x, |FinitePosterior.finiteStandardObservation spin T x v| ∂ρ) + (d : ℝ) * beta) ∧
0 ≤ (∫ t in 0..T, Q t) := by
classical
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have hcoords : ∀ (eta : D → IsingSpin) w, spinSign (eta w) = -1 ∨ spinSign (eta w) = 1 := by
intro eta w
cases eta w <;> simp [spinSign]
have h := FinitePosterior.integral_finiteStandardNormalizedEnergy_le
(μ.map D.restrict) (finiteIsingEnergy beta D)
(hμ.restrict_real_eq_normalized_exp D hD)
(fun (eta : D → IsingSpin) w => spinSign (eta w)) (fun eta => spinSign (eta v))
(fun eta => hcoords eta v) ⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩ hcoords hT
have hbound := hμ.integral_finiteIsingEntropy_le_own_observation hbeta D v T
simp only [finiteIsingEntropy_zero, sub_zero] at h
exact ⟨h.1, h.2.1.trans hbound, h.2.2⟩
end Problem367
end
end OAI
|