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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 | import OAI.Probability.FreeIsing.Ising.CopyCavityBounds
import OAI.Probability.FreeIsing.Ising.ObservationComparison
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal ENNReal
namespace Problem367.IsingCopyPathBounds
open IsingObservationInnovation IsingObservationFactor IsingCopyStationarity
variable {d : ℕ} {beta : ℝ}
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
def edgeLogDifference (hbeta : 0 ≤ beta) (t : ℝ≥0) (v w : TreeVertex d)
(q : IsingConditionalCopyBounds.CopySpace d) : ℝ :=
Real.log (responseAttenuation (Real.tanh beta)
(CavityArray.field (tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta)
(sitePair beta t q).1 v w)) -
Real.log (responseAttenuation (Real.tanh beta)
(CavityArray.field (tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta)
(sitePair beta t q).2 v w))
def edgeLogBound (d : ℕ) (beta : ℝ) (T : ℝ≥0) : ℝ :=
2 * ((2 * (T : ℝ) + 2 * (d : ℝ) * beta) / (1 - Real.tanh beta))
theorem edgeLogBound_nonneg (hbeta : 0 ≤ beta) (T : ℝ≥0) :
0 ≤ edgeLogBound d beta T := by
have hden : 0 < 1 - Real.tanh beta := sub_pos.mpr (Real.tanh_lt_one beta)
unfold edgeLogBound
positivity
theorem measurable_edgeLogDifference (hbeta : 0 ≤ beta) (t : ℝ≥0)
(v w : TreeVertex d) : Measurable (edgeLogDifference hbeta t v w) := by
have h0 := tanh_nonneg_of_beta_nonneg hbeta
have h1 := Real.tanh_lt_one beta
have hs : Real.tanh beta ^ 2 ≤ 1 := by nlinarith
have hl := (lipschitzWith_log_responseAttenuation hs).continuous.measurable
have hc := CavityArray.measurable_field_apply h0 h1 v w
exact (hl.comp (hc.comp (measurable_fst.comp (measurable_sitePair beta t)))).sub
(hl.comp (hc.comp (measurable_snd.comp (measurable_sitePair beta t))))
theorem ae_all_siteDifference_bound
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
{t T : ℝ≥0} (ht : t ≤ T) :
∀ᵐ q ∂copies beta μ, ∀ v : TreeVertex d,
|(sitePair beta t q).1 v - (sitePair beta t q).2 v| ≤
2 * (T : ℝ) + 2 * (d : ℝ) * beta := by
apply ae_all_iff.mpr
intro v
have : IsProbabilityMeasure ((observationLaw μ).map (innovationMap beta)) :=
inferInstance
have h := IsingConditionalCopyBounds.ae_fieldDifference_bound hμ hbeta
kernel_graphLaw t v
filter_upwards [h] with q hq
have ht' : (t : ℝ) ≤ (T : ℝ) := ht
exact hq.trans (by linarith)
theorem ae_edgeLogDifference_bound
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
{t T : ℝ≥0} (ht : t ≤ T) (v w : TreeVertex d) :
∀ᵐ q ∂copies beta μ,
|edgeLogDifference hbeta t v w q| ≤ edgeLogBound d beta T := by
have h0 := tanh_nonneg_of_beta_nonneg hbeta
have h1 := Real.tanh_lt_one beta
have hs : Real.tanh beta ^ 2 ≤ 1 := by nlinarith
filter_upwards [ae_all_siteDifference_bound hμ hbeta ht] with q hq
have hc := CavityArray.field_sub_abs_le h0 h1 v w (hq v) (hq w)
have hl := (lipschitzWith_log_responseAttenuation hs).dist_le_mul
(CavityArray.field h0 h1 (sitePair beta t q).1 v w)
(CavityArray.field h0 h1 (sitePair beta t q).2 v w)
apply (show |edgeLogDifference hbeta t v w q| ≤
2 * |CavityArray.field h0 h1 (sitePair beta t q).1 v w -
CavityArray.field h0 h1 (sitePair beta t q).2 v w| from by
simpa only [edgeLogDifference, Real.dist_eq, NNReal.coe_ofNat] using hl).trans
exact mul_le_mul_of_nonneg_left hc (by norm_num)
def pathLogDifference {ι : Type*} (hbeta : 0 ≤ beta) (t : ℝ≥0)
(s : Finset ι) (e : ι → TreeVertex d × TreeVertex d)
(q : IsingConditionalCopyBounds.CopySpace d) : ℝ :=
∑ i ∈ s, edgeLogDifference hbeta t (e i).1 (e i).2 q
theorem measurable_pathLogDifference {ι : Type*}
(hbeta : 0 ≤ beta) (t : ℝ≥0) (s : Finset ι)
(e : ι → TreeVertex d × TreeVertex d) :
Measurable (pathLogDifference hbeta t s e) :=
Finset.measurable_fun_sum s (fun i _ =>
measurable_edgeLogDifference hbeta t (e i).1 (e i).2)
theorem ae_pathLogDifference_bound {ι : Type*}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
{t T : ℝ≥0} (ht : t ≤ T) (s : Finset ι)
(e : ι → TreeVertex d × TreeVertex d) :
∀ᵐ q ∂copies beta μ,
|pathLogDifference hbeta t s e q| ≤ (s.card : ℝ) * edgeLogBound d beta T := by
classical
have hall : ∀ᵐ q ∂copies beta μ, ∀ i ∈ s,
|edgeLogDifference hbeta t (e i).1 (e i).2 q| ≤ edgeLogBound d beta T :=
(ae_ball_iff s.finite_toSet.countable).mpr (fun i _ =>
ae_edgeLogDifference_bound hμ hbeta ht (e i).1 (e i).2)
filter_upwards [hall] with q hq
calc
|pathLogDifference hbeta t s e q| ≤
∑ i ∈ s, |edgeLogDifference hbeta t (e i).1 (e i).2 q| :=
Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ _i ∈ s, edgeLogBound d beta T := Finset.sum_le_sum hq
_ = (s.card : ℝ) * edgeLogBound d beta T := by simp
theorem ae_pathLogDifference_sq_bound {ι : Type*}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
{t T : ℝ≥0} (ht : t ≤ T) (s : Finset ι)
(e : ι → TreeVertex d × TreeVertex d) :
∀ᵐ q ∂copies beta μ,
pathLogDifference hbeta t s e q ^ 2 ≤
(s.card : ℝ) ^ 2 * edgeLogBound d beta T ^ 2 := by
filter_upwards [ae_pathLogDifference_bound hμ hbeta ht s e] with q hq
have hb : 0 ≤ (s.card : ℝ) * edgeLogBound d beta T :=
mul_nonneg (Nat.cast_nonneg _) (edgeLogBound_nonneg hbeta T)
simpa only [sq_abs, mul_pow] using (sq_le_sq₀ (abs_nonneg _) hb).mpr hq
theorem memLp_pathLogDifference {ι : Type*}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (s : Finset ι) (e : ι → TreeVertex d × TreeVertex d)
(p : ℝ≥0∞) : MemLp (pathLogDifference hbeta t s e) p (copies beta μ) := by
apply MemLp.of_bound (measurable_pathLogDifference hbeta t s e).aestronglyMeasurable
((s.card : ℝ) * edgeLogBound d beta t)
simpa only [Real.norm_eq_abs] using ae_pathLogDifference_bound hμ hbeta le_rfl s e
theorem integrable_pathLogDifference_sq {ι : Type*}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (s : Finset ι) (e : ι → TreeVertex d × TreeVertex d) :
Integrable (fun q => pathLogDifference hbeta t s e q ^ 2) (copies beta μ) :=
(memLp_pathLogDifference hμ hbeta t s e 2).integrable_sq
end Problem367.IsingCopyPathBounds
end
end OAI
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