lean/OAI/Probability/InvariantIsing/Gaussian/GaussianBaseComparison.lean

import OAI.Probability.InvariantIsing.Gaussian.CountableGaussianComparison
import OAI.Probability.IsingPerceptron.MonomialGgAtMinimum
import OAI.Probability.IsingPerceptron.JointGGClosure

namespace OAI

/-! Stability of the actual Gaussian partition under bounded base-energy changes. -/

noncomputable section

open MeasureTheory ProbabilityTheory IsingPerceptron

namespace InvariantIsing

/-- An independent Gaussian perturbation changes the mean pressure by at
most half its maximal variance, including after a fixed base-energy tilt. -/
theorem countable_cylinder_log_mean_cost {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) :
    let d := (∫ g : ℕ → ℝ, Real.log (∫ x, Real.exp (H x + cylinderField (A x) g) ∂ν)
      ∂gaussianCoordinates) - Real.log (∫ x, Real.exp (H x) ∂ν)
    0 ≤ d ∧ d ≤ B / 2 := by
  have : IsProbabilityMeasure (ν.tilted H) := isProbabilityMeasure_tilted hH
  dsimp only
  rw [cylinder_log_partition_mean_base ν H hH A hA, add_sub_cancel_left]
  simpa only [one_pow, one_mul] using cylinder_cgf_mean_bounds (ν.tilted H) A hA 1

lemma cylinder_partition_exp_integrable_ae {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) :
    ∀ᵐ g : ℕ → ℝ ∂gaussianCoordinates,
      Integrable (fun x => Real.exp (H x + cylinderField (A x) g)) ν := by
  have : IsProbabilityMeasure (ν.tilted H) := isProbabilityMeasure_tilted hH
  have hv (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)) hv 1).1.prod_right_ae
  filter_upwards [he] with g hg
  simpa only [one_mul, smul_eq_mul, ← Real.exp_add] using
    (integrable_tilted_iff hH (fun x => Real.exp (1 * cylinderField (A x) g))).mp hg

/-- A uniform deterministic energy change bounds the mean logpartition
change, for every countable prior and bounded-variance Gaussian field. -/
theorem countable_cylinder_log_mean_base_compare {X : Type*} [MeasurableSpace X] [Countable X]
    [MeasurableSingletonClass X] (ν : Measure X) [IsProbabilityMeasure ν]
    (H J : X → ℝ) (hH : Integrable (fun x => Real.exp (H x)) ν)
    (hJ : Integrable (fun x => Real.exp (J x)) ν)
    (A : X → ℕ →₀ ℝ) {B K : ℝ} (hA : ∀ x, (A x).sum (fun _ c => c ^ 2) ≤ B)
    (hK : ∀ x, |H x - J x| ≤ K) :
    |(∫ g : ℕ → ℝ, Real.log (∫ x, Real.exp (H x + cylinderField (A x) g) ∂ν)
        ∂gaussianCoordinates) -
      ∫ g : ℕ → ℝ, Real.log (∫ x, Real.exp (J x + cylinderField (A x) g) ∂ν)
        ∂gaussianCoordinates| ≤ K := by
  have hiH := (cylinder_log_partition_memLp_two ν H hH A hA 1).integrable (by norm_num)
  have hiJ := (cylinder_log_partition_memLp_two ν J hJ A hA 1).integrable (by norm_num)
  simp only [one_mul] at hiH hiJ
  rw [← integral_sub hiH hiJ, ← Real.norm_eq_abs]
  apply (norm_integral_le_of_norm_le_const (μ := gaussianCoordinates) (C := K) ?_).trans_eq (by simp)
  filter_upwards [cylinder_partition_exp_integrable_ae ν H hH A hA,
    cylinder_partition_exp_integrable_ae ν J hJ A hA] with g hgH hgJ
  have h := logMean_abs_sub_le ν (b := 1) (by norm_num)
    (by simpa only [one_mul] using hgH) (by simpa only [one_mul] using hgJ)
    (fun x => by simpa only [add_sub_add_right_eq_sub] using hK x)
  simpa only [logMean, div_one, one_mul, Real.norm_eq_abs] using h

end InvariantIsing

end

end OAI