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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 | import OAI.Probability.FreeIsing.Conditioning.PlantedInformation
import OAI.Probability.FreeIsing.Energy.Attenuation
import OAI.Probability.FreeIsing.Posterior.InfinitePosteriorField
import OAI.Probability.FreeIsing.Posterior.SelectedPosteriorSymmetry
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367
open PlantedSymmetry
local instance componentInformationUpperNonemptyIsingSpin : Nonempty IsingSpin := ⟨.neg⟩
theorem singletonGibbsSpinMean_eq_tanh {d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
(v : TreeVertex d) (y : ({v} : Finset (TreeVertex d)) → ℝ) :
FinitePosterior.fieldMean (finiteIsingEnergy beta {v})
(fun (eta : ({v} : Finset (TreeVertex d)) → IsingSpin) w => spinSign (eta w))
(fun eta => spinSign (eta ⟨v, by simp⟩)) y = Real.tanh (y ⟨v, by simp⟩) := by
classical
let r : ({v} : Finset (TreeVertex d)) := ⟨v, by simp⟩
have hm := FinitePosterior.fieldMean_spin_mem_Ioo (finiteIsingEnergy beta {v})
(fun (eta : ({v} : Finset (TreeVertex d)) → IsingSpin) w => spinSign (eta w))
(fun eta => spinSign (eta r)) y
(fun eta => by cases eta r <;> simp [spinSign])
⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩
have hfield := finite_tree_gibbs_field_bound {v} hb y r
change |Real.artanh (FinitePosterior.fieldMean (finiteIsingEnergy beta {v})
(fun eta w => spinSign (eta w)) (fun eta => spinSign (eta r)) y) - y r| ≤ _ at hfield
simp only [inducedEdges_singleton, Finset.filter_empty, Finset.card_empty,
Nat.cast_zero, zero_mul] at hfield
have heq := sub_eq_zero.mp (abs_eq_zero.mp (le_antisymm hfield (abs_nonneg _)))
rw [← Real.tanh_artanh hm, heq]
theorem HasFreeIsingLaw.selected_singleton_gaussian_posterior
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(t : ℝ≥0) (v : TreeVertex d) :
selectedPosterior (μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(fun s z => (t : ℝ) * spinSign s + z) (fun _ : Unit => v) v =ᵐ[
μ.prod (gaussianNoiseLaw (TreeVertex d) t)]
fun p => Real.tanh (observation t p v) := by
classical
have hfield : MeasurableSpace.comap (finiteObservation {v} t) inferInstance =
selectedObservationField (fun s z => (t : ℝ) * spinSign s + z)
(fun _ : Unit => v) := by
change selectedObservationField (fun s z => (t : ℝ) * spinSign s + z)
(fun w : ({v} : Finset (TreeVertex d)) => w.1) = _
apply selectedObservationField_eq_of_range_eq
ext w
simp
have hce := hμ.finiteObservation_condExp {v} (connectedFinset_singleton v) t
(fun eta => spinSign (eta ⟨v, by simp⟩))
rw [hfield] at hce
filter_upwards [hce] with p hp
change selectedPosterior _ _ _ _ p = _ at hp
rw [hp, singletonGibbsSpinMean_eq_tanh hb]
rfl
theorem gaussian_observation_second_moment {d : ℕ}
(μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ]
(t : ℝ≥0) (v : TreeVertex d) :
(∫ p, (observation t p v) ^ 2 ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) =
(t : ℝ) + (t : ℝ) ^ 2 := by
let ρ := μ.prod (gaussianNoiseLaw (TreeVertex d) t)
have hS : MemLp (fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) =>
spinSign (p.1 v)) 2 ρ :=
(PlantedInformation.spinSign_memLp μ v).comp_measurePreserving measurePreserving_fst
have hB := centered_gaussian_moments (PlantedInformation.noise_hasLaw μ t v)
have hind := PlantedInformation.spin_noise_indep μ t v v
have h := integral_observation_mul_observation hS hS hB.1 hB.1 hind hind
hB.2.1 hB.2.1 (t : ℝ)
have hspin : (∫ p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ),
spinSign (p.1 v) * spinSign (p.1 v) ∂ρ) = 1 := by
have heq : (fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) =>
spinSign (p.1 v) * spinSign (p.1 v)) = fun _ => (1 : ℝ) := by
funext p
cases p.1 v <;> norm_num [spinSign]
rw [heq]
simp [ρ]
rw [hspin] at h
simp only [← pow_two] at h
rw [hB.2.2] at h
simpa only [observation, pow_two, mul_one, add_comm] using h
theorem HasFreeIsingLaw.selected_singleton_information_le
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(t : ℝ≥0) (v : TreeVertex d) :
(∫ p, (selectedPosterior (μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(fun s z => (t : ℝ) * spinSign s + z) (fun _ : Unit => v) v p) ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ (t : ℝ) + (t : ℝ) ^ 2 := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
let ρ := μ.prod (gaussianNoiseLaw (TreeVertex d) t)
have hF : Measurable (Function.uncurry
(fun (s : IsingSpin) (z : ℝ) => (t : ℝ) * spinSign s + z)) :=
((PlantedInformation.measurable_spinSign.comp measurable_fst).const_mul _).add measurable_snd
have hpost := integrable_selectedPosterior_abs_pow ρ hF (fun _ : Unit => v) v 2
simp only [sq_abs] at hpost
have hS := (PlantedInformation.spinSign_memLp μ v).comp_measurePreserving
(measurePreserving_fst (ν := gaussianNoiseLaw (TreeVertex d) t))
have hB := (centered_gaussian_moments (PlantedInformation.noise_hasLaw μ t v)).1
have hobs : Integrable (fun p => (observation t p v) ^ 2) ρ :=
((hS.const_mul (t : ℝ)).add hB).integrable_sq
rw [← gaussian_observation_second_moment μ t v]
apply integral_mono_ae hpost hobs
filter_upwards [hμ.selected_singleton_gaussian_posterior hb t v] with p hp
rw [hp]
have hbound := abs_tanh_le_abs_response (observation t p v)
simpa only [sq_abs] using pow_le_pow_left₀ (abs_nonneg _) hbound 2
end Problem367
end
end OAI
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