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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 | import OAI.Probability.FreeIsing.Cavity.MomentTransport
import OAI.Probability.FreeIsing.Conditioning.SelectedPosteriorEvenConditioning
import OAI.Probability.FreeIsing.Energy.SelectedCriticalFieldMoments
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367
open PlantedSymmetry
theorem HasFreeIsingLaw.selectedGaussianField_abs_identDistrib_cond_spin
{d : ℕ} {ι : Type*} {theta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)} (hμ : HasFreeIsingLaw theta μ)
(t : ℝ≥0) (v : ι → TreeVertex d) (r w : TreeVertex d) (s : IsingSpin) :
IdentDistrib (fun p => |selectedGaussianField μ t v r p|)
(fun p => |selectedGaussianField μ t v r p|)
((μ.prod (gaussianNoiseLaw (TreeVertex d) t))[|{p | p.1 w = s}])
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) :=
hμ.selected_gaussian_field_abs_identDistrib_cond_spin t t v r w s
theorem HasFreeIsingLaw.integrable_selectedGaussianField_cond_abs_pow
{d : ℕ} {ι : Type*} {beta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(t : ℝ≥0) (v : ι → TreeVertex d) (r : TreeVertex d)
(hr : r ∈ Set.range v) (w : TreeVertex d) (s : IsingSpin) (p : ℕ) :
Integrable (fun q => |selectedGaussianField μ t v r q| ^ p)
((μ.prod (gaussianNoiseLaw (TreeVertex d) t))[|{q | q.1 w = s}]) := by
have h := (hμ.selectedGaussianField_abs_identDistrib_cond_spin t v r w s).comp
(measurable_id.pow_const p)
exact h.integrable_iff.mpr (hμ.integrable_selectedGaussianField_abs_pow hb t v r hr p)
theorem HasFreeIsingLaw.childComponentField_cond_abs_identDistrib
{d : ℕ} {theta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw theta μ) (t : ℝ≥0) (c a : Fin d)
(w : TreeVertex d) (s : IsingSpin) :
IdentDistrib
(fun p => |selectedGaussianField μ t (childComponentIndexVertex c a)
(prefixFlip c (RootedBroadcast.branchRoot a)) p|)
(fun p => |selectedGaussianField μ t (componentIndexVertex (c := c))
(RootedBroadcast.branchRoot c) p|)
((μ.prod (gaussianNoiseLaw (TreeVertex d) t))[|{p | p.1 w = s}])
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) := by
have hF : Measurable (Function.uncurry
(fun (a : IsingSpin) (z : ℝ) => (t : ℝ) * spinSign a + z)) :=
((PlantedInformation.measurable_spinSign.comp measurable_fst).const_mul _).add
measurable_snd
have ha : Measurable (fun x : ℝ => |Real.artanh x|) := by
unfold Real.artanh
fun_prop
apply (hμ.selectedGaussianField_abs_identDistrib_cond_spin t
(childComponentIndexVertex c a) (prefixFlip c (RootedBroadcast.branchRoot a)) w s).trans
exact (childComponentPosterior_identDistrib hμ (gaussianReal 0 t) hF c a).comp ha
theorem HasFreeIsingLaw.childComponentField_cond_response_moment
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(t : ℝ≥0) (c a : Fin d) (w : TreeVertex d) (s : IsingSpin) (p : ℕ) :
Integrable
(fun q => |cavityMessage (Real.tanh beta)
(selectedGaussianField μ t (childComponentIndexVertex c a)
(prefixFlip c (RootedBroadcast.branchRoot a)) q)| ^ p)
((μ.prod (gaussianNoiseLaw (TreeVertex d) t))[|{q | q.1 w = s}]) ∧
(∫ q, |cavityMessage (Real.tanh beta)
(selectedGaussianField μ t (childComponentIndexVertex c a)
(prefixFlip c (RootedBroadcast.branchRoot a)) q)| ^ p
∂((μ.prod (gaussianNoiseLaw (TreeVertex d) t))[|{q | q.1 w = s}])) ≤
(Real.tanh beta) ^ p * ∫ q,
|selectedGaussianField μ t (componentIndexVertex (c := c))
(RootedBroadcast.branchRoot c) q| ^ p
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t) := by
have h0 := tanh_nonneg_of_beta_nonneg hb
have h1 := (Real.tanh_lt_one beta).le
have hr : RootedBroadcast.branchRoot c ∈ Set.range (componentIndexVertex (c := c)) := by
apply mem_range_componentIndexVertex.mpr
exact ⟨[], by simp [RootedBroadcast.branchRoot]⟩
exact response_abs_moment_of_identDistrib_abs
(hμ.childComponentField_cond_abs_identDistrib t c a w s)
(lipschitzWith_cavityMessage h0 h1).continuous.measurable
(measurable_selectedGaussianField μ t _ _)
(cavityMessage_abs_le_mul h0 h1) p
(hμ.integrable_selectedGaussianField_abs_pow hb t _ _ hr p)
end Problem367
end
end OAI
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