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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 | import OAI.Probability.FreeIsing.Energy.CoupledAttenuation
import OAI.Probability.FreeIsing.Energy.MomentTail
import OAI.Probability.FreeIsing.Estimates.LogInterpolation
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
def cavityLogVarianceCoefficient (theta : ℝ) : ℝ :=
4 / (1 - theta) ^ 2
def cavityLogTailCoefficient (theta M : ℝ) : ℝ :=
8 * (M / (1 - theta)) ^ 2
theorem cavityLogVarianceCoefficient_nonneg (theta : ℝ) :
0 ≤ cavityLogVarianceCoefficient theta := by
unfold cavityLogVarianceCoefficient
positivity
theorem cavityLogTailCoefficient_nonneg (theta M : ℝ) :
0 ≤ cavityLogTailCoefficient theta M := by
unfold cavityLogTailCoefficient
positivity
section Coupling
variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
variable {theta : ℝ} {X Y X' Y' H K H' K' : Ω → ℝ}
theorem integral_cavity_log_discrepancy_sq_le_of_marginal_tails
(hθ0 : 0 ≤ theta) (hθ1 : theta < 1)
(hX : Measurable X) (hX' : Measurable X')
(hH : Integrable (fun ω => (H ω - H' ω) ^ 2) μ)
(hK : Integrable (fun ω => (K ω - K' ω) ^ 2) μ)
(heq : ∀ᵐ ω ∂μ,
X ω + cavityMessage theta (Y ω) = H ω ∧
Y ω + cavityMessage theta (X ω) = K ω ∧
X' ω + cavityMessage theta (Y' ω) = H' ω ∧
Y' ω + cavityMessage theta (X' ω) = K' ω)
{V M R B₁ B₂ : ℝ} (hM : 0 ≤ M)
(hbound : ∀ᵐ ω ∂μ, |H ω - H' ω| ≤ M ∧ |K ω - K' ω| ≤ M)
(hHV : (∫ ω, (H ω - H' ω) ^ 2 ∂μ) ≤ V)
(hKV : (∫ ω, (K ω - K' ω) ^ 2 ∂μ) ≤ V)
(hTail : μ.real {ω | R < |X ω|} ≤ B₁)
(hTail' : μ.real {ω | R < |X' ω|} ≤ B₂) :
(∫ ω, (Real.log (responseAttenuation theta (X ω)) -
Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤
cavityLogVarianceCoefficient theta * R ^ 2 * V +
min (cavityLogVarianceCoefficient theta * V)
(4 * (M / (1 - theta)) ^ 2 * (B₁ + B₂)) := by
have hθsq : theta ^ 2 ≤ 1 := by nlinarith
have hInt := integrable_cavity_pair_sub_sq hθ0 hθ1 hX hX' hH hK heq
have hVar := integral_cavity_pair_sub_sq_le hθ0 hθ1 hX hX' hH hK heq hHV hKV
have hBound : ∀ᵐ ω ∂μ, |X ω - X' ω| ≤ M / (1 - theta) := by
filter_upwards [heq, hbound] with ω he hb
exact cavity_pair_sub_abs_le hθ0 hθ1 he.1 he.2.1 he.2.2.1 he.2.2.2 hb.1 hb.2
have h := integral_log_responseAttenuation_sub_sq_le_of_marginal_tails
hθsq hX hX' hInt (div_nonneg hM (by linarith)) hBound hVar hTail hTail'
have hfirst : 4 * R ^ 2 * (V / (1 - theta) ^ 2) =
cavityLogVarianceCoefficient theta * R ^ 2 * V := by
unfold cavityLogVarianceCoefficient
ring
have hsecond : 4 * (V / (1 - theta) ^ 2) =
cavityLogVarianceCoefficient theta * V := by
unfold cavityLogVarianceCoefficient
ring
simpa only [hfirst, hsecond] using h
theorem integral_cavity_log_discrepancy_small_time
(hθ0 : 0 ≤ theta) (hθ1 : theta < 1)
(hX : Measurable X) (hX' : Measurable X')
(hH : Integrable (fun ω => (H ω - H' ω) ^ 2) μ)
(hK : Integrable (fun ω => (K ω - K' ω) ^ 2) μ)
(heq : ∀ᵐ ω ∂μ,
X ω + cavityMessage theta (Y ω) = H ω ∧
Y ω + cavityMessage theta (X ω) = K ω ∧
X' ω + cavityMessage theta (Y' ω) = H' ω ∧
Y' ω + cavityMessage theta (X' ω) = K' ω)
{V M t c : ℝ} (hM : 0 ≤ M) (ht : 0 < t)
(hbound : ∀ᵐ ω ∂μ, |H ω - H' ω| ≤ M ∧ |K ω - K' ω| ≤ M)
(hHV : (∫ ω, (H ω - H' ω) ^ 2 ∂μ) ≤ V)
(hKV : (∫ ω, (K ω - K' ω) ^ 2 ∂μ) ≤ V)
(hTail : μ.real {ω | t ^ (1 / 8 : ℝ) < |X ω|} ≤
Real.exp (-c * (Real.log (1 / t)) ^ 2))
(hTail' : μ.real {ω | t ^ (1 / 8 : ℝ) < |X' ω|} ≤
Real.exp (-c * (Real.log (1 / t)) ^ 2)) :
(∫ ω, (Real.log (responseAttenuation theta (X ω)) -
Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤
cavityLogVarianceCoefficient theta * t ^ (1 / 4 : ℝ) * V +
min (cavityLogVarianceCoefficient theta * V)
(cavityLogTailCoefficient theta M * Real.exp (-c * (Real.log (1 / t)) ^ 2)) := by
have h := integral_cavity_log_discrepancy_sq_le_of_marginal_tails
hθ0 hθ1 hX hX' hH hK heq hM hbound hHV hKV hTail hTail'
have hr : (t ^ (1 / 8 : ℝ)) ^ 2 = t ^ (1 / 4 : ℝ) := by
rw [← Real.rpow_natCast, ← Real.rpow_mul ht.le]
norm_num
have htail : 4 * (M / (1 - theta)) ^ 2 *
(Real.exp (-c * (Real.log (1 / t)) ^ 2) + Real.exp (-c * (Real.log (1 / t)) ^ 2)) =
cavityLogTailCoefficient theta M * Real.exp (-c * (Real.log (1 / t)) ^ 2) := by
unfold cavityLogTailCoefficient
ring
simpa only [hr, htail] using h
theorem integral_cavity_log_discrepancy_small_time_of_moments
(hθ0 : 0 ≤ theta) (hθ1 : theta < 1)
(hX : Measurable X) (hX' : Measurable X')
(hH : Integrable (fun ω => (H ω - H' ω) ^ 2) μ)
(hK : Integrable (fun ω => (K ω - K' ω) ^ 2) μ)
(heq : ∀ᵐ ω ∂μ,
X ω + cavityMessage theta (Y ω) = H ω ∧
Y ω + cavityMessage theta (X ω) = K ω ∧
X' ω + cavityMessage theta (Y' ω) = H' ω ∧
Y' ω + cavityMessage theta (X' ω) = K' ω)
{V M t A : ℝ} (hM : 0 ≤ M) (ht : 0 < t) (hA : 0 < A)
(hsmall : 32 * A ≤ Real.log (1 / t))
(hbound : ∀ᵐ ω ∂μ, |H ω - H' ω| ≤ M ∧ |K ω - K' ω| ≤ M)
(hHV : (∫ ω, (H ω - H' ω) ^ 2 ∂μ) ≤ V)
(hKV : (∫ ω, (K ω - K' ω) ^ 2 ∂μ) ≤ V)
(hInt : ∀ p : ℕ, 1 ≤ p → Integrable (fun ω => |X ω| ^ p) μ)
(hInt' : ∀ p : ℕ, 1 ≤ p → Integrable (fun ω => |X' ω| ^ p) μ)
(hmom : ∀ p : ℕ, 1 ≤ p → (∫ ω, |X ω| ^ p ∂μ) ≤
Real.exp (A * (p : ℝ) ^ 2) * t ^ ((p : ℝ) / 4))
(hmom' : ∀ p : ℕ, 1 ≤ p → (∫ ω, |X' ω| ^ p ∂μ) ≤
Real.exp (A * (p : ℝ) ^ 2) * t ^ ((p : ℝ) / 4)) :
(∫ ω, (Real.log (responseAttenuation theta (X ω)) -
Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤
cavityLogVarianceCoefficient theta * t ^ (1 / 4 : ℝ) * V +
min (cavityLogVarianceCoefficient theta * V)
(cavityLogTailCoefficient theta M *
Real.exp (-(1 / (512 * A)) * (Real.log (1 / t)) ^ 2)) := by
have he : -(1 / (512 * A)) * (Real.log (1 / t)) ^ 2 =
-((Real.log (1 / t)) ^ 2) / (512 * A) := by ring
apply integral_cavity_log_discrepancy_small_time hθ0 hθ1 hX hX' hH hK heq
hM ht hbound hHV hKV
· rw [he]
exact measureReal_abs_gt_rpow_le_of_moments hA ht hsmall hInt hmom
· rw [he]
exact measureReal_abs_gt_rpow_le_of_moments hA ht hsmall hInt' hmom'
theorem integral_cavity_log_discrepancy_small_time_of_common_law
{Ξ : Type*} [MeasurableSpace Ξ] {ν : Measure Ξ} {Z : Ξ → ℝ}
(hθ0 : 0 ≤ theta) (hθ1 : theta < 1)
(hX : Measurable X) (hX' : Measurable X')
(hH : Integrable (fun ω => (H ω - H' ω) ^ 2) μ)
(hK : Integrable (fun ω => (K ω - K' ω) ^ 2) μ)
(heq : ∀ᵐ ω ∂μ,
X ω + cavityMessage theta (Y ω) = H ω ∧
Y ω + cavityMessage theta (X ω) = K ω ∧
X' ω + cavityMessage theta (Y' ω) = H' ω ∧
Y' ω + cavityMessage theta (X' ω) = K' ω)
{V M t c : ℝ} (hM : 0 ≤ M) (ht : 0 < t)
(hbound : ∀ᵐ ω ∂μ, |H ω - H' ω| ≤ M ∧ |K ω - K' ω| ≤ M)
(hHV : (∫ ω, (H ω - H' ω) ^ 2 ∂μ) ≤ V)
(hKV : (∫ ω, (K ω - K' ω) ^ 2 ∂μ) ≤ V)
(hLaw : IdentDistrib X Z μ ν) (hLaw' : IdentDistrib X' Z μ ν)
(hTail : ν {z | t ^ (1 / 8 : ℝ) < |Z z|} ≤
ENNReal.ofReal (Real.exp (-c * (Real.log (1 / t)) ^ 2))) :
(∫ ω, (Real.log (responseAttenuation theta (X ω)) -
Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤
cavityLogVarianceCoefficient theta * t ^ (1 / 4 : ℝ) * V +
min (cavityLogVarianceCoefficient theta * V)
(cavityLogTailCoefficient theta M * Real.exp (-c * (Real.log (1 / t)) ^ 2)) := by
have transfer {W : Ω → ℝ} (hW : IdentDistrib W Z μ ν) :
μ.real {ω | t ^ (1 / 8 : ℝ) < |W ω|} ≤
Real.exp (-c * (Real.log (1 / t)) ^ 2) := by
apply ENNReal.toReal_le_of_le_ofReal (Real.exp_nonneg _)
have hm := (hW.comp measurable_abs).measure_mem_eq
(measurableSet_Ioi : MeasurableSet (Set.Ioi (t ^ (1 / 8 : ℝ))))
change μ {ω | t ^ (1 / 8 : ℝ) < |W ω|} =
ν {z | t ^ (1 / 8 : ℝ) < |Z z|} at hm
rw [hm]
exact hTail
exact integral_cavity_log_discrepancy_small_time hθ0 hθ1 hX hX' hH hK heq
hM ht hbound hHV hKV (transfer hLaw) (transfer hLaw')
omit [IsProbabilityMeasure μ] in
theorem integral_cavity_log_discrepancy_osgood_of_small_time
{V M t c L : ℝ} (hc : 0 < c) (hL : 1 ≤ L)
(hV : 0 < V) (hVL : V ≤ L) (ht : 0 < t) (ht1 : t < 1)
(hsmall : (∫ ω, (Real.log (responseAttenuation theta (X ω)) -
Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤
cavityLogVarianceCoefficient theta * t ^ (1 / 4 : ℝ) * V +
min (cavityLogVarianceCoefficient theta * V)
(cavityLogTailCoefficient theta M * Real.exp (-c * (Real.log (1 / t)) ^ 2))) :
(∫ ω, (Real.log (responseAttenuation theta (X ω)) -
Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤
(cavityLogVarianceCoefficient theta +
2 * (cavityLogVarianceCoefficient theta + cavityLogTailCoefficient theta M) / c) *
V * (t ^ (1 / 4 : ℝ) + (1 + Real.log (L / V)) / (Real.log (1 / t)) ^ 2) := by
exact small_time_log_loss_of_minimum (cavityLogVarianceCoefficient_nonneg theta)
(cavityLogTailCoefficient_nonneg theta M) hc hL hV hVL ht ht1 hsmall
end Coupling
end Problem367
end
end OAI
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