lean/OAI/Probability/InvariantIsing/Gaussian/BaseGaussianVariance.lean

import OAI.Probability.InvariantIsing.Gaussian.CountableGaussianVariance

namespace OAI

/-! Gaussian variance for the countable spin/leaf partition with base energy. -/

noncomputable section

open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped NNReal

namespace InvariantIsing

lemma cylinder_cgf_memLp_two {X : Type*} [MeasurableSpace X] [Countable X]
    [MeasurableSingletonClass X] (ν : Measure X) [IsProbabilityMeasure ν]
    (A : X → ℕ →₀ ℝ) {B : ℝ} (hA : ∀ x, (A x).sum (fun _ c => c ^ 2) ≤ B)
    (t : ℝ) :
    MemLp (fun g : ℕ → ℝ => cgf (fun x => cylinderField (A x) g) ν t)
      2 gaussianCoordinates := by
  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)
  exact (memLp_two_iff_integrable_sq
    hH.exp.stronglyMeasurable.integral_prod_right'.measurable.log.aestronglyMeasurable).mpr
    (gaussian_log_partition_second (ν := ν) hH hLaw hVar).1

/-- Splitting off a fixed integrable base energy is an almost-everywhere
identity for the actual countable Gaussian partition, including nonuniform
and cascade-leaf weights. -/
lemma cylinder_log_partition_base {X : Type*} [MeasurableSpace X] [Countable X]
    [MeasurableSingletonClass X] (ν : Measure X) [IsProbabilityMeasure ν]
    (H : X → ℝ) (hH : Integrable (fun x => Real.exp (H x)) ν)
    (A : X → ℕ →₀ ℝ) {B : ℝ} (hA : ∀ x, (A x).sum (fun _ c => c ^ 2) ≤ B) (t : ℝ) :
    (fun g : ℕ → ℝ => Real.log (∫ x, Real.exp (H x + t * cylinderField (A x) g) ∂ν))
      =ᵐ[gaussianCoordinates]
    (fun g => Real.log (∫ x, Real.exp (H x) ∂ν) +
      cgf (fun x => cylinderField (A x) g) (ν.tilted H) t) := by
  have : IsProbabilityMeasure (ν.tilted H) := isProbabilityMeasure_tilted hH
  have hVar (x : X) : (((A x).sum (fun _ c => c ^ 2)).toNNReal : ℝ) ≤ B := by
    rw [Real.coe_toNNReal _ (show 0 ≤ (A x).sum (fun _ c => c ^ 2) from
      Finset.sum_nonneg (fun _ _ => sq_nonneg _))]
    exact hA x
  have he := (gaussian_field_exp_integrable (ν := ν.tilted H)
    (measurable_cylinderFields A) (fun x => cylinderField_law (A x)) hVar t).1.prod_right_ae
  filter_upwards [he] with g hg
  have hz : 0 < ∫ x, Real.exp (H x) ∂ν := integral_exp_pos hH
  have hz' : 0 < ∫ x, Real.exp (t * cylinderField (A x) g) ∂ν.tilted H := integral_exp_pos hg
  have hi := integral_exp_tilted (μ := ν) H (fun x => t * cylinderField (A x) g)
  have hid : (∫ x, Real.exp (H x + t * cylinderField (A x) g) ∂ν) =
      (∫ x, Real.exp (H x) ∂ν) *
        (∫ x, Real.exp (t * cylinderField (A x) g) ∂ν.tilted H) := by
    rw [hi]
    simp only [Pi.add_apply]
    field_simp
  rw [hid, Real.log_mul hz.ne' hz'.ne']
  rfl

lemma cylinder_log_partition_memLp_two {X : Type*} [MeasurableSpace X] [Countable X]
    [MeasurableSingletonClass X] (ν : Measure X) [IsProbabilityMeasure ν]
    (H : X → ℝ) (hH : Integrable (fun x => Real.exp (H x)) ν)
    (A : X → ℕ →₀ ℝ) {B : ℝ} (hA : ∀ x, (A x).sum (fun _ c => c ^ 2) ≤ B) (t : ℝ) :
    MemLp (fun g : ℕ → ℝ => Real.log
      (∫ x, Real.exp (H x + t * cylinderField (A x) g) ∂ν)) 2 gaussianCoordinates := by
  have : IsProbabilityMeasure (ν.tilted H) := isProbabilityMeasure_tilted hH
  have he := cylinder_log_partition_base ν H hH A hA t
  exact MemLp.ae_eq (Filter.EventuallyEq.symm he)
    ((memLp_const (Real.log (∫ x, Real.exp (H x) ∂ν))).add
      (cylinder_cgf_memLp_two (ν.tilted H) A hA t))

/-- The actual countable Gaussian partition with arbitrary integrable base
energy has the same coefficient variance bound. This is conditional only on
the specified published Gaussian Poincaré consequence. -/
theorem countable_cylinder_log_partition_variance_le (hpub : GaussianLipschitzVarianceInput)
    {X : Type*} [MeasurableSpace X] [Countable X] [MeasurableSingletonClass X]
    (ν : Measure X) [IsProbabilityMeasure ν] (H : X → ℝ)
    (hH : Integrable (fun x => Real.exp (H x)) 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 => Real.log
      (∫ x, Real.exp (H x + t * cylinderField (A x) g) ∂ν)) gaussianCoordinates ≤ t ^ 2 * B := by
  have : IsProbabilityMeasure (ν.tilted H) := isProbabilityMeasure_tilted hH
  have he := cylinder_log_partition_base ν H hH A hA t
  rw [variance_congr he, variance_const_add (cylinder_cgf_memLp_two (ν.tilted H) A hA t).aestronglyMeasurable]
  exact countable_cylinder_cgf_variance_le hpub (ν.tilted H) A B hB hA t

end InvariantIsing

end

end OAI