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a89d282 | 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 | import OAI.Probability.InvariantIsing.Pressure.VarianceLimit
import OAI.Probability.InvariantIsing.Core.FiniteGaussianVariance
import OAI.Probability.IsingPerceptron.NormalizedRestrictionPi
namespace OAI
/-! The variance estimate passes from finite positive-mass leaf cutoffs to the
actual countable spin/cascade-leaf prior. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter
open scoped Topology NNReal
namespace InvariantIsing
theorem cylinder_cgf_variance_le_of_finite_restrictions {X : Type*}
[MeasurableSpace X] [Countable X] [MeasurableSingletonClass X]
(ν : Measure X) [IsProbabilityMeasure ν] (A : X → ℕ →₀ ℝ)
{B C : ℝ} (hA : ∀ x, (A x).sum (fun _ c => c ^ 2) ≤ B) (t : ℝ)
(hfinite : ∀ S : Set X, S.Finite → ν S ≠ 0 →
variance (fun g : ℕ → ℝ => cgf (fun x => cylinderField (A x) g)
(normalizedRestriction ν S) t) gaussianCoordinates ≤ C) :
variance (fun g : ℕ → ℝ => cgf (fun x => cylinderField (A x) g) ν t)
gaussianCoordinates ≤ C := by
obtain ⟨S, hS, hpos, hExh⟩ := countable_reference_exhaustion ν
let νn := fun n => normalizedRestriction ν (S n)
have : ∀ n, IsProbabilityMeasure (νn n) := fun n => normalizedRestriction_probability (hpos n)
let H := fun p : (ℕ → ℝ) × X => t * cylinderField (A p.2) p.1
have hH : Measurable H := (measurable_cylinderFields A).const_mul t
have hLaw (x : X) : gaussianCoordinates.map (fun g => H (g, x)) =
gaussianReal 0 (⟨t ^ 2, sq_nonneg t⟩ * (((A x).sum (fun _ c => c ^ 2)).toNNReal)) :=
cylinder_scaled_law A x t
have hVar (x : X) :
((⟨t ^ 2, sq_nonneg t⟩ : ℝ≥0) * (((A x).sum (fun _ c => c ^ 2)).toNNReal) : ℝ) ≤ t ^ 2 * B := by
change t ^ 2 * (((A x).sum (fun _ c => c ^ 2)).toNNReal : ℝ) ≤ _
rw [Real.coe_toNNReal _ (show 0 ≤ (A x).sum (fun _ c => c ^ 2) from
Finset.sum_nonneg (fun _ _ => sq_nonneg _))]
exact mul_le_mul_of_nonneg_left (hA x) (sq_nonneg t)
let f := fun n g => Real.log (∫ x, Real.exp (H (g, x)) ∂νn n)
let F := fun g => Real.log (∫ x, Real.exp (H (g, x)) ∂ν)
have hfm (n : ℕ) : Measurable (f n) :=
hH.exp.stronglyMeasurable.integral_prod_right'.measurable.log
have hFm : Measurable F := hH.exp.stronglyMeasurable.integral_prod_right'.measurable.log
have hf (n : ℕ) : MemLp (f n) 2 gaussianCoordinates :=
(memLp_two_iff_integrable_sq (hfm n).aestronglyMeasurable).mpr
(gaussian_log_partition_second (ν := νn n) hH hLaw hVar).1
have hF : MemLp F 2 gaussianCoordinates :=
(memLp_two_iff_integrable_sq hFm.aestronglyMeasurable).mpr
(gaussian_log_partition_second (ν := ν) hH hLaw hVar).1
have hm : Tendsto (fun n => ∫ g, f n g ∂gaussianCoordinates) atTop
(𝓝 (∫ g, F g ∂gaussianCoordinates)) :=
gaussian_log_restriction_expectation_tendsto hH hLaw hVar
(fun n => (hS n).measurableSet) hExh hpos
have he := (gaussian_field_exp_integrable (ν := ν) hH hLaw hVar 1).1
simp only [one_mul] at he
have ht : ∀ᵐ g ∂gaussianCoordinates, Tendsto (fun n => f n g) atTop (𝓝 (F g)) := by
filter_upwards [he.prod_right_ae] with g hg
have hz : 0 < ∫ x, Real.exp (H (g, x)) ∂ν := MeasureTheory.integral_exp_pos hg
exact (Real.continuousAt_log hz.ne').tendsto.comp
(normalized_restriction_integral_tendsto (fun n => (hS n).measurableSet) hExh hg)
have hv (n : ℕ) : variance (f n) gaussianCoordinates ≤ C :=
hfinite (S n) (hS n) (hpos n)
exact variance_le_of_ae_tendsto_of_mean_tendsto hf hF ht hm hv
/-- The countable spin/leaf Gaussian log partition inherits the uniform
coefficient variance bound from the finite-dimensional Gaussian
Poincaré inequality. -/
theorem countable_cylinder_cgf_variance_le (hpub : GaussianLipschitzVarianceInput)
{X : Type*} [MeasurableSpace X] [Countable X] [MeasurableSingletonClass X]
(ν : Measure X) [IsProbabilityMeasure ν] (A : X → ℕ →₀ ℝ)
(B : ℝ) (hB : 0 ≤ B) (hA : ∀ x, (A x).sum (fun _ c => c ^ 2) ≤ B) (t : ℝ) :
variance (fun g : ℕ → ℝ => cgf (fun x => cylinderField (A x) g) ν t)
gaussianCoordinates ≤ t ^ 2 * B := by
apply cylinder_cgf_variance_le_of_finite_restrictions ν A hA t
intro S hS hpos
let := hS.fintype
have : IsProbabilityMeasure (subtypeReference ν S) :=
subtypeReference_probability ν hS.measurableSet hpos
have h := finite_cylinder_cgf_variance_le hpub (subtypeReference ν S)
(fun x : S => A x) B hB (fun x => hA x) t
have he (g : ℕ → ℝ) :
cgf (fun x : S => cylinderField (A x) g) (subtypeReference ν S) t =
cgf (fun x => cylinderField (A x) g) (normalizedRestriction ν S) t := by
unfold cgf mgf
congr 1
exact (subtypeReference_preserving ν hS.measurableSet).hasLaw.integral_comp
(measurable_of_countable (fun x => Real.exp (t * cylinderField (A x) g))).aestronglyMeasurable
simpa only [he] using h
end InvariantIsing
end
end OAI
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