lean/OAI/Probability/Perceptron/Variational/CountableGaussianIBP.lean

import Mathlib
import OAI.Probability.Perceptron.Variational.CountableGaussianResampling
import OAI.Probability.Perceptron.Interpolation.IntegrableReplicaGibbs

namespace OAI

noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology NNReal ENNReal BigOperators
namespace SphericalPerceptronFreeEnergy
variable {S : Type*} [MeasurableSpace S] (μ : Measure S) [IsProbabilityMeasure μ]

omit [IsProbabilityMeasure μ] in
lemma countableGaussian_affine_zero_exp_integrable (i : ℕ)
    {H : (ℕ → ℝ) → S → ℝ} {v : S → ℝ} (hv : Measurable v)
    {C : ℝ} (hvC : ∀ x, |v x| ≤ C)
    (haff : ∀ g t x, H (Function.update g i t) x = H (Function.update g i 0) x+t*v x)
    (hi : ∀ᵐ g ∂countableGaussianLaw, Integrable (fun x => Real.exp (H g x)) μ) :
    ∀ᵐ g ∂countableGaussianLaw,
      Integrable (fun x => Real.exp (H (Function.update g i 0) x)) μ := by
  have hp := (resampleGaussianCoordinate_measurePreserving i).quasiMeasurePreserving.ae hi
  have hs := Measure.measurePreserving_swap.quasiMeasurePreserving.ae hp
  filter_upwards [Measure.ae_ae_of_ae_prod hs] with g hg
  obtain ⟨t,ht⟩ := hg.exists
  change Integrable (fun x => Real.exp (H (Function.update g i t) x)) μ at ht
  have hb (x) : ‖Real.exp (-t*v x)‖ ≤ Real.exp (|t| * C) := by
    rw [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)]
    apply Real.exp_le_exp.mpr
    exact (le_abs_self _).trans (by
      rw [abs_mul,abs_neg]
      exact mul_le_mul_of_nonneg_left (hvC x) (abs_nonneg t))
  have hh := ht.mul_bdd ((hv.const_mul (-t)).exp.aestronglyMeasurable)
    (ae_of_all _ hb)
  apply hh.congr
  exact ae_of_all _ fun x => by
    dsimp only
    rw [← Real.exp_add,haff]
    congr 1
    ring

lemma countableGaussian_coordinate_tilt_ibp (i : ℕ)
    {H : (ℕ → ℝ) → S → ℝ} {v F : S → ℝ}
    (hH : Measurable (Function.uncurry H)) (hv : Measurable v) (hF : Measurable F)
    {C D : ℝ} (hC : 0 ≤ C) (hD : 0 ≤ D)
    (hvC : ∀ x, |v x| ≤ C) (hFD : ∀ x, |F x| ≤ D)
    (haff : ∀ g t x, H (Function.update g i t) x = H (Function.update g i 0) x+t*v x)
    (hi0 : ∀ᵐ g ∂countableGaussianLaw,
      Integrable (fun x => Real.exp (H (Function.update g i 0) x)) μ) :
    (∫ g, g i*tiltMean μ (H g) F 1 ∂countableGaussianLaw) =
      ∫ g, tiltMean μ (H g) (fun x => v x*F x) 1-
        tiltMean μ (H g) F 1*tiltMean μ (H g) v 1 ∂countableGaussianLaw := by
  let R (g : ℕ → ℝ) := tiltMean μ (H g) (fun x => v x*F x) 1-
    tiltMean μ (H g) F 1*tiltMean μ (H g) v 1
  have hm (K : S → ℝ) (hK : Measurable K) := measurable_tiltMean μ hH hK
  have hHg (g) : Measurable (H g) := hH.comp (measurable_const.prodMk measurable_id)
  have hb (K : S → ℝ) (hK : Measurable K) {B : ℝ} (hB : 0 ≤ B)
      (hKB : ∀ x, |K x| ≤ B) (g) : |tiltMean μ (H g) K 1| ≤ B :=
    tiltMean_bound_general μ (hHg g) hK hB hKB
  have hvF (x : S) : |v x*F x| ≤ C*D := by
    rw [abs_mul]
    exact mul_le_mul (hvC x) (hFD x) (abs_nonneg _) hC
  have hRm : Measurable R := (hm _ (hv.mul hF)).sub ((hm _ hF).mul (hm _ hv))
  have hRb (g) : |R g| ≤ 2*C*D := by
    calc
      _ ≤ |tiltMean μ (H g) (fun x => v x*F x) 1|+
          |tiltMean μ (H g) F 1*tiltMean μ (H g) v 1| := abs_sub _ _
      _ ≤ C*D+D*C := add_le_add (hb _ (hv.mul hF) (mul_nonneg hC hD) hvF g) (by
        rw [abs_mul]
        exact mul_le_mul (hb _ hF hD hFD g) (hb _ hv hC hvC g) (abs_nonneg _) hD)
      _ = _ := by ring
  let e := resampleGaussianCoordinate i
  have he := resampleGaussianCoordinate_measurePreserving i
  have intmap (K : (ℕ → ℝ) → ℝ) (hK : Measurable K) :
      (∫ p, K (e p) ∂(gaussianReal 0 1).prod countableGaussianLaw) = ∫ g, K g ∂countableGaussianLaw := by
    simpa only [he.map_eq] using
      (integral_map (μ := (gaussianReal 0 1).prod countableGaussianLaw) he.measurable.aemeasurable hK.aestronglyMeasurable).symm
  have hmF : Measurable (fun p : ℝ×(ℕ → ℝ) => tiltMean μ (H (e p)) F 1) :=
    (hm _ hF).comp he.measurable
  have hiF : Integrable (fun p : ℝ×(ℕ → ℝ) => p.1*tiltMean μ (H (e p)) F 1)
      ((gaussianReal 0 1).prod countableGaussianLaw) := by
    apply ((memLp_id_gaussianReal (μ := 0) (v := 1) 1).integrable (by norm_num)).comp_fst countableGaussianLaw |>.mul_bdd
      hmF.aestronglyMeasurable
    exact ae_of_all _ fun p => by simpa only [Real.norm_eq_abs] using hb _ hF hD hFD (e p)
  have hiR : Integrable (fun p => R (e p)) ((gaussianReal 0 1).prod countableGaussianLaw) :=
    Integrable.of_bound (hRm.comp he.measurable).aestronglyMeasurable (2*C*D)
      (ae_of_all _ fun p => by simpa only [Real.norm_eq_abs] using hRb (e p))
  change (∫ g, g i*tiltMean μ (H g) F 1 ∂countableGaussianLaw) = ∫ g, R g ∂countableGaussianLaw
  rw [← intmap (fun g => g i*tiltMean μ (H g) F 1) ((measurable_pi_apply i).mul (hm _ hF)),← intmap _ hRm]
  simp only [e,resampleGaussianCoordinate,Function.update_self]
  change (∫ p, p.1*tiltMean μ (H (e p)) F 1 ∂(gaussianReal 0 1).prod countableGaussianLaw) =
    ∫ p, R (e p) ∂(gaussianReal 0 1).prod countableGaussianLaw
  rw [integral_prod_symm _ hiF,integral_prod_symm _ hiR]
  apply integral_congr_ae
  filter_upwards [hi0] with g hg
  let w : S → ℝ := H (Function.update g i 0)
  have hwm : Measurable w := hHg _
  have hwI : Integrable (fun x => Real.exp (1*w x)) μ := by simpa only [one_mul] using hg
  let := tilt_law_probability_of_integrable μ hwI
  have heK (K : S → ℝ) (u : ℝ) : tiltMean (tiltLaw μ w 1) v K u =
      tiltMean μ (H (Function.update g i u)) K 1 := by
    rw [tilt_mean_twice_of_integrable μ hwm hwI u]
    congr 2
    funext x
    simp only [one_mul]
    exact (haff g u x).symm
  have hh := gaussian_tilt_integrationByParts (tiltLaw μ w 1) hv hF hC hD hvC hFD
  simp_rw [heK] at hh
  exact hh

lemma countableGaussian_coordinate_tilt_ibp_of_exp_integrable (i : ℕ)
    {H : (ℕ → ℝ) → S → ℝ} {v F : S → ℝ}
    (hH : Measurable (Function.uncurry H)) (hv : Measurable v) (hF : Measurable F)
    {C D : ℝ} (hC : 0 ≤ C) (hD : 0 ≤ D)
    (hvC : ∀ x, |v x| ≤ C) (hFD : ∀ x, |F x| ≤ D)
    (haff : ∀ g t x, H (Function.update g i t) x = H (Function.update g i 0) x+t*v x)
    (hi : ∀ᵐ g ∂countableGaussianLaw, Integrable (fun x => Real.exp (H g x)) μ) :
    (∫ g, g i*tiltMean μ (H g) F 1 ∂countableGaussianLaw) =
      ∫ g, tiltMean μ (H g) (fun x => v x*F x) 1-
        tiltMean μ (H g) F 1*tiltMean μ (H g) v 1 ∂countableGaussianLaw :=
  countableGaussian_coordinate_tilt_ibp μ i hH hv hF hC hD hvC hFD haff
    (countableGaussian_affine_zero_exp_integrable μ i hv hvC haff hi)

end SphericalPerceptronFreeEnergy
end

end OAI