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import OAI.Computability.VertexCover.Games.Stochastic
namespace OAI
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namespace UniqueGames.Foundations.Games.FiniteDistribution
open scoped BigOperators
noncomputable section
theorem iid_pushforward {Ω Γ : Type*} [Fintype Ω] [Fintype Γ]
(μ : FiniteDistribution Ω) (f : Ω → Γ) (n : Nat) :
(μ.pushforward f).iid n =
(μ.iid n).pushforward (fun answers i => f (answers i)) := by
classical
apply eq_of_weight_eq
intro y
simp only [iid, pushforward]
rw [Fintype.prod_sum]
apply Finset.sum_congr rfl
intro x _
by_cases h : (fun i => f (x i)) = y
· have hp : ∀ i, f (x i) = y i := congrFun h
simp [hp]
· rw [ite_eq_right h]
have hn : ¬ ∀ i, f (x i) = y i := fun hp => h (funext hp)
obtain ⟨i, hi⟩ := not_forall.mp hn
apply Finset.prod_eq_zero (Finset.mem_univ i)
simp [hi]
def uniform (Ω : Type*) [Fintype Ω] [Nonempty Ω] : FiniteDistribution Ω where
weight _ := 1 / (Fintype.card Ω : ℝ)
nonnegative _ := div_nonneg zero_le_one (Nat.cast_nonneg _)
normalized := by
have hn : (Fintype.card Ω : ℝ) ≠ 0 :=
Nat.cast_ne_zero.mpr Fintype.card_ne_zero
simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, one_div]
exact mul_inv_cancel₀ hn
theorem iid_uniform {Ω : Type*} [Fintype Ω] [Nonempty Ω] (n : Nat) :
(uniform Ω).iid n = uniform (Fin n → Ω) := by
classical
apply eq_of_weight_eq
intro x
simp [iid, uniform, Nat.cast_pow, one_div]
theorem expectation_uniform {Ω : Type*} [Fintype Ω] [Nonempty Ω] (f : Ω → ℝ) :
(uniform Ω).expectation f = (∑ x, f x) / (Fintype.card Ω : ℝ) := by
unfold expectation uniform
rw [← Finset.mul_sum]
simp [div_eq_mul_inv, mul_comm]
end
end UniqueGames.Foundations.Games.FiniteDistribution
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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