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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
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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