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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 | import OAI.Probability.FreeIsing.Posterior.VarianceInverse
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.FinitePosteriorVariance
open PlantedSymmetry
local instance : Nonempty IsingSpin := ⟨.neg⟩
def spinMean {d : ℕ} (beta : ℝ) (D : Finset (TreeVertex d)) (v : D)
(y : TreeVertex d → ℝ) : ℝ :=
FinitePosterior.fieldMean (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) w => spinSign (eta w))
(fun eta => spinSign (eta v)) (D.restrict y)
theorem continuous_spinMean {d : ℕ} (beta : ℝ)
(D : Finset (TreeVertex d)) (v : D) : Continuous (spinMean beta D v) := by
apply (FinitePosterior.continuous_fieldMean _ _ _).comp
exact continuous_pi (fun w => continuous_apply w.1)
theorem spinMean_mem_Ioo {d : ℕ} (beta : ℝ)
(D : Finset (TreeVertex d)) (v : D) (y : TreeVertex d → ℝ) :
spinMean beta D v y ∈ Set.Ioo (-1 : ℝ) 1 := by
exact FinitePosterior.fieldMean_spin_mem_Ioo _ _ _ _
(fun eta => by cases eta v <;> simp [spinSign])
⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩
theorem field_sub_observation_abs_le {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta)
(D : Finset (TreeVertex d)) (v : D) (y : TreeVertex d → ℝ) :
|Real.artanh (spinMean beta D v y) - y v| ≤ (d : ℝ) * beta := by
exact finite_tree_gibbs_field_uniform_bound D hbeta (D.restrict y) v
theorem field_abs_le_noise {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta)
(D : Finset (TreeVertex d)) (v : D) (t : ℝ≥0)
(p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) :
|Real.artanh (spinMean beta D v (observation t p))| ≤
(d : ℝ) * beta + (t : ℝ) + |p.2 v| := by
have ho : |observation t p v| ≤ (t : ℝ) + |p.2 v| := by
calc
_ ≤ |(t : ℝ) * spinSign (p.1 v)| + |p.2 v| := abs_add_le _ _
_ = _ := by
cases hs : p.1 v <;> simp [spinSign, abs_of_nonneg t.coe_nonneg]
calc
_ ≤ |Real.artanh (spinMean beta D v (observation t p)) - observation t p v| +
|observation t p v| := by
simpa using abs_sub_le (Real.artanh (spinMean beta D v (observation t p)))
(observation t p v) 0
_ ≤ (d : ℝ) * beta + ((t : ℝ) + |p.2 v|) :=
add_le_add (field_sub_observation_abs_le hbeta D v _) ho
_ = _ := by ring
private theorem measurable_field {d : ℕ} (beta : ℝ)
(D : Finset (TreeVertex d)) (v : D) (t : ℝ≥0) :
Measurable (fun p => Real.artanh (spinMean beta D v (observation t p))) := by
have ha : Measurable Real.artanh := by unfold Real.artanh; fun_prop
exact ha.comp ((continuous_spinMean beta D v).measurable.comp (measurable_observation t))
theorem integrable_exp_abs_field {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta)
(μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ]
(D : Finset (TreeVertex d)) (v : D) (t : ℝ≥0) {a : ℝ} (ha : 0 ≤ a) :
Integrable (fun p => Real.exp (a * |Real.artanh
(spinMean beta D v (observation t p))|))
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) := by
exact integrable_exp_abs_of_gaussian_bound (measurable_field beta D v t)
(PlantedInformation.noise_hasLaw μ t v) ha
(Filter.Eventually.of_forall (field_abs_le_noise hbeta D v t))
theorem integral_exp_abs_field_le {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta)
(μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ]
(D : Finset (TreeVertex d)) (v : D) (t : ℝ≥0) {a : ℝ} (ha : 0 ≤ a) :
(∫ p, Real.exp (a * |Real.artanh (spinMean beta D v (observation t p))|)
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤
2 * Real.exp (a * ((d : ℝ) * beta + t) + (t : ℝ) * a ^ 2 / 2) := by
exact integral_exp_abs_le_of_gaussian_bound (measurable_field beta D v t)
(PlantedInformation.noise_hasLaw μ t v) ha
(Filter.Eventually.of_forall (field_abs_le_noise hbeta D v t))
theorem inverse_variance_pow_le_noise {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta)
(D : Finset (TreeVertex d)) (v : D) (t : ℝ≥0) (k : ℕ)
(p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) :
((1 - (spinMean beta D v (observation t p)) ^ 2)⁻¹) ^ k ≤
Real.exp ((2 * (k : ℝ)) * ((d : ℝ) * beta + (t : ℝ) + |p.2 v|)) := by
apply (inverse_spin_variance_pow_le_exp (spinMean_mem_Ioo beta D v _) k).trans
exact Real.exp_le_exp.mpr
(mul_le_mul_of_nonneg_left (field_abs_le_noise hbeta D v t p) (by positivity))
theorem integrable_inverse_variance_pow {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta)
(μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ]
(D : Finset (TreeVertex d)) (v : D) (t : ℝ≥0) (k : ℕ) :
Integrable (fun p => ((1 - (spinMean beta D v (observation t p)) ^ 2)⁻¹) ^ k)
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) := by
apply (integrable_exp_abs_field hbeta μ D v t
(a := 2 * (k : ℝ)) (by positivity)).mono'
· exact ((measurable_const.sub (((continuous_spinMean beta D v).measurable.comp
(measurable_observation t)).pow_const 2)).inv.pow_const k).aestronglyMeasurable
· apply Filter.Eventually.of_forall
intro p
have hm := spinMean_mem_Ioo beta D v (observation t p)
have hpos : 0 < 1 - (spinMean beta D v (observation t p)) ^ 2 := by
nlinarith [mul_pos (sub_pos.mpr hm.2) (show 0 < 1 + spinMean beta D v
(observation t p) by linarith [hm.1])]
rw [Real.norm_of_nonneg (pow_nonneg (inv_nonneg.mpr hpos.le) _)]
exact inverse_spin_variance_pow_le_exp hm k
theorem integral_inverse_variance_pow_le {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta)
(μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ]
(D : Finset (TreeVertex d)) (v : D) (t : ℝ≥0) (k : ℕ) :
(∫ p, ((1 - (spinMean beta D v (observation t p)) ^ 2)⁻¹) ^ k
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤
2 * Real.exp ((2 * (k : ℝ)) * ((d : ℝ) * beta + t) +
(t : ℝ) * (2 * (k : ℝ)) ^ 2 / 2) := by
calc
_ ≤ ∫ p, Real.exp ((2 * (k : ℝ)) * |Real.artanh
(spinMean beta D v (observation t p))|)
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t) := by
apply integral_mono (integrable_inverse_variance_pow hbeta μ D v t k)
(integrable_exp_abs_field hbeta μ D v t (by positivity))
intro p
exact inverse_spin_variance_pow_le_exp (spinMean_mem_Ioo beta D v _) k
_ ≤ _ := integral_exp_abs_field_le hbeta μ D v t (by positivity)
theorem condExp_eq_spinMean {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≥0) :
(μ.prod (gaussianNoiseLaw (TreeVertex d) t))[
fun p => spinSign (p.1 v) | MeasurableSpace.comap (finiteObservation D t) inferInstance]
=ᵐ[μ.prod (gaussianNoiseLaw (TreeVertex d) t)]
fun p => spinMean beta D v (observation t p) :=
hμ.finiteObservation_condExp D hD t (fun eta => spinSign (eta v))
end Problem367.FinitePosteriorVariance
end
end OAI
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