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namespace OAI
/-!
# Moments of products on a subset of coordinates
Under the actual product law, observables supported on a finite subset
factor into their coordinate expectations. Extending the factors by one
outside the subset reduces this to the full product moment identity.
-/
open scoped BigOperators
noncomputable section
namespace TalagrandThreshold.FiniteLaw
variable {I : Type*} [Fintype I] [DecidableEq I]
{A : I → Type*} [∀ i, Fintype (A i)]
/-- A product on a subset has the product of its actual coordinate means. -/
theorem expectation_pi_subset_product (μ : ∀ i, FiniteLaw (A i))
(S : Finset I) (f : ∀ i, A i → ℝ) :
(pi μ).expectation (fun ω => ∏ i ∈ S, f i (ω i)) =
∏ i ∈ S, (μ i).expectation (f i) := by
classical
let g : ∀ i, A i → ℝ := fun i a => if i ∈ S then f i a else 1
have hg (ω : ∀ i, A i) : (∏ i, g i (ω i)) = ∏ i ∈ S, f i (ω i) := by
simpa only [g] using Fintype.prod_ite_mem S (fun i => f i (ω i))
have he (i : I) : (μ i).expectation (g i) =
if i ∈ S then (μ i).expectation (f i) else 1 := by
by_cases hi : i ∈ S
· simp only [g, ite_eq_left hi]
· simp only [g, ite_eq_right hi, expectation_const]
calc
_ = (pi μ).expectation (fun ω => ∏ i, g i (ω i)) := by
congr 1
funext ω
exact (hg ω).symm
_ = ∏ i, (μ i).expectation (g i) := expectation_pi_product μ g
_ = ∏ i ∈ S, (μ i).expectation (f i) := by
simp only [he, Finset.prod_ite_mem_eq]
/-- Equal coordinate means give the corresponding power of the subset size. -/
theorem expectation_pi_subset_product_of_mean (μ : ∀ i, FiniteLaw (A i))
(S : Finset I) (f : ∀ i, A i → ℝ) (a : ℝ)
(hmean : ∀ i ∈ S, (μ i).expectation (f i) = a) :
(pi μ).expectation (fun ω => ∏ i ∈ S, f i (ω i)) = a ^ S.card := by
rw [expectation_pi_subset_product]
calc
(∏ i ∈ S, (μ i).expectation (f i)) = ∏ _i ∈ S, a :=
Finset.prod_congr rfl hmean
_ = a ^ S.card := Finset.prod_const a
/-- Identically distributed coordinates give a power of their common mean. -/
theorem expectation_pi_subset_product_const {Ω : Type*} [Fintype Ω]
(μ : FiniteLaw Ω) (S : Finset I) (f : Ω → ℝ) :
(pi (fun _ : I => μ)).expectation (fun ω => ∏ i ∈ S, f (ω i)) =
(μ.expectation f) ^ S.card := by
rw [expectation_pi_subset_product, Finset.prod_const]
end TalagrandThreshold.FiniteLaw
end
end OAI
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