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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 | import OAI.Probability.FreeIsing.Estimates.MessageFunction
import OAI.Probability.FreeIsing.Ising.ConditionalCopyBounds
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped NNReal
namespace Problem367
theorem abs_artanh_le_two_mul_of_abs_le_half {x : ℝ} (hx : |x| ≤ 1 / 2) :
|Real.artanh x| ≤ 2 * |x| := by
have hd (z : ℝ) (hz : z ∈ Icc (-1 / 2 : ℝ) (1 / 2)) :
HasDerivWithinAt Real.artanh (1 / (1 - z ^ 2)) (Icc (-1 / 2) (1 / 2)) z :=
(hasDerivAt_artanh_interval (by constructor <;> linarith [hz.1, hz.2])).hasDerivWithinAt
have hb (z : ℝ) (hz : z ∈ Icc (-1 / 2 : ℝ) (1 / 2)) :
‖1 / (1 - z ^ 2)‖ ≤ (2 : ℝ) := by
have hsq : z ^ 2 ≤ 1 / 4 := by
nlinarith [mul_nonneg (by linarith [hz.1] : 0 ≤ z + 1 / 2)
(by linarith [hz.2] : 0 ≤ 1 / 2 - z)]
have hden : 0 < 1 - z ^ 2 := by linarith
rw [Real.norm_eq_abs, abs_of_nonneg (by positivity : 0 ≤ 1 / (1 - z ^ 2))]
exact (div_le_iff₀ hden).mpr (by linarith)
have h := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hd hb (convex_Icc _ _)
(by norm_num : (0 : ℝ) ∈ Icc (-1 / 2 : ℝ) (1 / 2))
(show x ∈ Icc (-1 / 2 : ℝ) (1 / 2) by
constructor <;> linarith [(abs_le.mp hx).1, (abs_le.mp hx).2])
simpa only [Real.artanh_zero, sub_zero, Real.norm_eq_abs] using h
theorem artanh_sub_sq_le_information {x y K : ℝ}
(hK : |Real.artanh x - Real.artanh y| ≤ K) :
(Real.artanh x - Real.artanh y) ^ 2 ≤ (8 + 4 * K ^ 2) * (x ^ 2 + y ^ 2) := by
have hK0 : 0 ≤ K := (abs_nonneg _).trans hK
have hdiff : (Real.artanh x - Real.artanh y) ^ 2 ≤ K ^ 2 := by
simpa only [sq_abs] using (pow_le_pow_left₀ (abs_nonneg _) hK 2)
by_cases hx : |x| ≤ 1 / 2
· by_cases hy : |y| ≤ 1 / 2
· have hax : (Real.artanh x) ^ 2 ≤ 4 * x ^ 2 := by
have h := mul_self_le_mul_self (abs_nonneg _) (abs_artanh_le_two_mul_of_abs_le_half hx)
nlinarith only [h, sq_abs (Real.artanh x), sq_abs x]
have hay : (Real.artanh y) ^ 2 ≤ 4 * y ^ 2 := by
have h := mul_self_le_mul_self (abs_nonneg _) (abs_artanh_le_two_mul_of_abs_le_half hy)
nlinarith only [h, sq_abs (Real.artanh y), sq_abs y]
have hsum : 0 ≤ 4 * K ^ 2 * (x ^ 2 + y ^ 2) := by positivity
nlinarith [sq_nonneg (Real.artanh x + Real.artanh y)]
· have hy' : 1 / 2 ≤ |y| := (lt_of_not_ge hy).le
have hy2 := mul_self_le_mul_self (by norm_num : (0 : ℝ) ≤ 1 / 2) hy'
have hmul := mul_le_mul_of_nonneg_left hy2 (sq_nonneg K)
nlinarith [sq_abs y, sq_nonneg x, sq_nonneg y, mul_nonneg (sq_nonneg K) (sq_nonneg x)]
· have hx' : 1 / 2 ≤ |x| := (lt_of_not_ge hx).le
have hx2 := mul_self_le_mul_self (by norm_num : (0 : ℝ) ≤ 1 / 2) hx'
have hmul := mul_le_mul_of_nonneg_left hx2 (sq_nonneg K)
nlinarith [sq_abs x, sq_nonneg x, sq_nonneg y, mul_nonneg (sq_nonneg K) (sq_nonneg y)]
namespace IsingConditionalCopyBounds
open IsingObservationInnovation
def posteriorInformation {d : ℕ} (beta : ℝ) (μ : Measure (TreeVertex d → IsingSpin))
(v : TreeVertex d) (t : ℝ≥0) : ℝ :=
∫ x, (IsingObservationInnovation.posterior beta v t x) ^ 2 ∂observationLaw μ
theorem posteriorInformation_eq_gaussian
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (t : ℝ≥0) (v : TreeVertex d) :
posteriorInformation beta μ v t =
∫ p, (PlantedSymmetry.posteriorMagnetization (μ.prod (PlantedSymmetry.gaussianNoiseLaw
(TreeVertex d) t)) t v p) ^ 2 ∂μ.prod
(PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have heval := hasLaw_observationLaw_eval μ t
have hobs : HasLaw (PlantedSymmetry.observation t)
((μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)).map
(PlantedSymmetry.observation t))
(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) :=
⟨(PlantedSymmetry.measurable_observation t).aemeasurable, rfl⟩
have hsame := (heval.identDistrib hobs).comp
((measurable_infiniteGibbsSpinMean beta v).pow_const 2)
calc
_ = ∫ p, (infiniteGibbsSpinMean beta v (PlantedSymmetry.observation t p)) ^ 2
∂μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t) := hsame.integral_eq
_ = _ := integral_congr_ae
((hμ.infiniteGibbsSpinMean_eq_posterior t v).fun_comp (fun z : ℝ => z ^ 2))
theorem fieldDifference_secondMoment_le_information
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
{rho : Measure (Paths d)} [IsProbabilityMeasure rho]
{eta : Kernel (Paths d) (Paths d)} [IsMarkovKernel eta]
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(hjoint : rho ⊗ₘ eta = graphLaw beta μ) (t : ℝ≥0) (v : TreeVertex d) :
(∫ q, (fieldDifference beta v t q) ^ 2 ∂rho ⊗ₘ (eta ×ₖ eta)) ≤
2 * (8 + 4 * (2 * (t : ℝ) + 2 * (d : ℝ) * beta) ^ 2) *
posteriorInformation beta μ v t := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
let : IsProbabilityMeasure (labelMeasure d) := by unfold labelMeasure; infer_instance
let : IsProbabilityMeasure (observationLaw μ) := by
unfold observationLaw
infer_instance
let m : Paths d → ℝ := IsingObservationInnovation.posterior beta v t
let C : ℝ := 8 + 4 * (2 * (t : ℝ) + 2 * (d : ℝ) * beta) ^ 2
have hm : Measurable m :=
(((posterior_progressive beta v).stronglyAdapted t).mono ((filtration d).le t)).measurable
have hmLp : MemLp m 2 (observationLaw μ) :=
MemLp.of_bound hm.aestronglyMeasurable 1
(Filter.Eventually.of_forall (fun x => posterior_norm_le_one beta v t x))
obtain ⟨hl, hr⟩ := measurePreserving_observation_copies hjoint
have hml := hmLp.comp_measurePreserving hl
have hmr := hmLp.comp_measurePreserving hr
have hdiff := (fieldDifference_secondMoment_bounds hμ hbeta hjoint t v).1
have hlint := hl.hasLaw.integral_comp (f := fun x => (m x) ^ 2)
(hm.pow_const 2).aestronglyMeasurable
have hrint := hr.hasLaw.integral_comp (f := fun x => (m x) ^ 2)
(hm.pow_const 2).aestronglyMeasurable
simp only [Function.comp_def] at hlint hrint
calc
(∫ q, (fieldDifference beta v t q) ^ 2 ∂rho ⊗ₘ (eta ×ₖ eta)) ≤
∫ q : CopySpace d, C * ((m q.2.1) ^ 2 + (m q.2.2) ^ 2)
∂rho ⊗ₘ (eta ×ₖ eta) := by
apply integral_mono_ae hdiff.integrable_sq
((hml.integrable_sq.add hmr.integrable_sq).const_mul C)
filter_upwards [ae_fieldDifference_bound hμ hbeta hjoint t v] with q hq
exact artanh_sub_sq_le_information hq
_ = C * ((∫ q : CopySpace d, (m q.2.1) ^ 2 ∂rho ⊗ₘ (eta ×ₖ eta)) +
∫ q : CopySpace d, (m q.2.2) ^ 2 ∂rho ⊗ₘ (eta ×ₖ eta)) := by
rw [integral_const_mul]
congr 1
exact integral_add hml.integrable_sq hmr.integrable_sq
_ = 2 * (8 + 4 * (2 * (t : ℝ) + 2 * (d : ℝ) * beta) ^ 2) *
posteriorInformation beta μ v t := by
rw [hlint, hrint]
change C * (posteriorInformation beta μ v t + posteriorInformation beta μ v t) = _
dsimp [C]
ring
theorem fieldDifference_secondMoment_le_horizon_information
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
{rho : Measure (Paths d)} [IsProbabilityMeasure rho]
{eta : Kernel (Paths d) (Paths d)} [IsMarkovKernel eta]
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(hjoint : rho ⊗ₘ eta = graphLaw beta μ) {T t : ℝ≥0} (ht : t ≤ T)
(v : TreeVertex d) :
(∫ q, (fieldDifference beta v t q) ^ 2 ∂rho ⊗ₘ (eta ×ₖ eta)) ≤
2 * (8 + 4 * (2 * (T : ℝ) + 2 * (d : ℝ) * beta) ^ 2) *
posteriorInformation beta μ v t := by
apply (fieldDifference_secondMoment_le_information hμ hbeta hjoint t v).trans
have hI : 0 ≤ posteriorInformation beta μ v t := integral_nonneg (fun _ => sq_nonneg _)
apply mul_le_mul_of_nonneg_right _ hI
have htR : (t : ℝ) ≤ (T : ℝ) := ht
have hK : 0 ≤ 2 * (t : ℝ) + 2 * (d : ℝ) * beta := by positivity
have hsq := pow_le_pow_left₀ hK
(show 2 * (t : ℝ) + 2 * (d : ℝ) * beta ≤ 2 * (T : ℝ) + 2 * (d : ℝ) * beta by linarith) 2
linarith
end IsingConditionalCopyBounds
end Problem367
end
end OAI
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