lean/OAI/Computability/VertexCover/Analysis/VariancePi.lean

import Mathlib
import OAI.Computability.VertexCover.Analysis.VarianceProduct

namespace OAI

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namespace VertexCover.Product
open MeasureTheory ProbabilityTheory
open scoped ENNReal

variable {X : Type*} [MeasurableSpace X]
variable (μ : Measure X) [IsProbabilityMeasure μ]

theorem cons_measurePreserving (n : ℕ) :
    MeasurePreserving (fun p : X × (Fin n → X) => Fin.cons p.1 p.2)
      (μ.prod (Measure.pi (fun _ : Fin n => μ))) (Measure.pi (fun _ : Fin (n+1) => μ)) := by
  simpa [MeasurableEquiv.piFinSuccAbove_symm_apply, Fin.insertNthEquiv,
    Fin.insertNth_zero', Fin.zero_succAbove] using
    (measurePreserving_piFinSuccAbove (fun _ : Fin (n+1) => μ) 0).symm

theorem integral_cons (n : ℕ) (f : (Fin (n+1) → X) → ℝ)
    (hi : Integrable f (Measure.pi (fun _ => μ))) :
    ∫ v, f v ∂Measure.pi (fun _ : Fin (n+1) => μ) =
      ∫ x, ∫ s, f (Fin.cons x s) ∂Measure.pi (fun _ : Fin n => μ) ∂μ := by
  have hp := cons_measurePreserving μ n
  calc
    _ = ∫ p : X × (Fin n → X), f (Fin.cons p.1 p.2) ∂μ.prod (Measure.pi (fun _ => μ)) := by
      rw [← hp.map_eq]
      exact integral_map hp.aemeasurable (by rw [hp.map_eq]; exact hi.aestronglyMeasurable)
    _ = _ := integral_prod _ (hp.integrable_comp_of_integrable hi)

noncomputable def coordinateVariance {n : ℕ} (f : (Fin n → X) → ℝ)
    (j : Fin n) (s : Fin n → X) : ℝ :=
  variance (fun x => f (Function.update s j x)) μ

theorem update_joint_measurable {n : ℕ} (j : Fin n) :
    Measurable (fun p : (Fin n → X) × X => Function.update p.1 j p.2) := by
  apply Measurable.of_eval
  intro k
  by_cases hk : k = j
  · subst k; simpa using measurable_snd
  · simpa only [Function.update_of_ne hk, Function.comp_def] using (measurable_pi_apply k).comp measurable_fst

theorem coordinateVariance_integrable {n : ℕ} (f : (Fin n → X) → ℝ)
    (hf : Measurable f) {C : ℝ} (hb : ∀ s, ‖f s‖ ≤ C) (j : Fin n) :
    Integrable (coordinateVariance μ f j) (Measure.pi (fun _ : Fin n => μ)) :=
  section_variance_integrable _ μ _ (hf.comp (update_joint_measurable j)) (fun _s _x => hb _)

theorem variance_pi_le (n : ℕ) (f : (Fin n → X) → ℝ) (hf : Measurable f)
    {C : ℝ} (hb : ∀ s, ‖f s‖ ≤ C) :
    variance f (Measure.pi (fun _ : Fin n => μ)) ≤
      ∑ j, ∫ s, coordinateVariance μ f j s ∂Measure.pi (fun _ : Fin n => μ) := by
  induction n with
  | zero =>
      have he : f = fun _ => f (fun i => Fin.elim0 i) := by
        funext s
        exact congrArg f (Subsingleton.elim _ _)
      rw [he, variance_eq_integral measurable_const.aemeasurable]
      simp
  | succ n ih =>
      let P := Measure.pi (fun _ : Fin n => μ)
      let F : X → (Fin n → X) → ℝ := fun x s => f (Fin.cons x s)
      have hF : Measurable (Function.uncurry F) := hf.comp (cons_measurePreserving μ n).measurable
      have hFb : ∀ x s, ‖F x s‖ ≤ C := fun x s => hb _
      have hsec : ∀ x, Measurable (F x) := fun x => hF.comp (measurable_const.prodMk measurable_id)
      have hpre : variance f (Measure.pi (fun _ : Fin (n+1) => μ)) ≤
          (∫ x, variance (F x) P ∂μ) + ∫ s, variance (fun x => F x s) μ ∂P := by
        rw [← (cons_measurePreserving μ n).variance_fun_comp hf.aemeasurable]
        exact variance_product_le μ P F hF hFb
      have hci := fun j : Fin (n+1) => coordinateVariance_integrable μ f hf hb j
      have hstep : ∀ x s (j : Fin n),
          coordinateVariance μ (F x) j s = coordinateVariance μ f j.succ (Fin.cons x s) := by
        intro x s j
        unfold coordinateVariance F
        simp only [Fin.cons_update]
      have hIi : ∀ j : Fin n, Integrable
          (fun x => ∫ s, coordinateVariance μ (F x) j s ∂P) μ := by
        intro j
        simp_rw [hstep]
        exact ((cons_measurePreserving μ n).integrable_comp_of_integrable (hci j.succ)).integral_prod_left
      have ht : (∫ x, variance (F x) P ∂μ) ≤
          ∑ j : Fin n, ∫ s, coordinateVariance μ f j.succ s ∂Measure.pi (fun _ => μ) := by
        calc
          _ ≤ ∫ x, ∑ j : Fin n, ∫ s, coordinateVariance μ (F x) j s ∂P ∂μ := by
            apply integral_mono (section_variance_integrable μ P F hF hFb)
              (integrable_finsetSum _ (fun j _ => hIi j))
            intro x
            exact ih (F x) (hsec x) (hFb x)
          _ = _ := by
            rw [integral_finsetSum _ (fun j _ => hIi j)]
            apply Finset.sum_congr rfl
            intro j hj
            simp_rw [hstep]
            exact (integral_cons μ n _ (hci j.succ)).symm
      have hz : (∫ s, variance (fun x => F x s) μ ∂P) =
          ∫ s, coordinateVariance μ f 0 s ∂Measure.pi (fun _ : Fin (n+1) => μ) := by
        rw [integral_cons μ n _ (hci 0)]
        simp only [coordinateVariance, Fin.update_cons_zero]
        simp only [F, integral_const, probReal_univ, one_smul]
        rfl
      calc
        _ ≤ (∫ x, variance (F x) P ∂μ) + ∫ s, variance (fun x => F x s) μ ∂P := hpre
        _ ≤ (∑ j : Fin n, ∫ s, coordinateVariance μ f j.succ s ∂Measure.pi (fun _ => μ)) +
            ∫ s, coordinateVariance μ f 0 s ∂Measure.pi (fun _ => μ) := by
          rw [← hz]
          exact add_le_add ht le_rfl
        _ = _ := by rw [Fin.sum_univ_succ]; exact add_comm _ _

end VertexCover.Product


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end OAI