| import OAI.Combinatorics.ExpectationThreshold.ProbabilityProduct |
|
|
| 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 |
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| end OAI |
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|
