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import OAI.Computability.VertexCover.Games.Basic
import OAI.Computability.VertexCover.Information.Finite
namespace OAI
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namespace UniqueGames.Foundations.Games.FiniteDistribution
open scoped BigOperators
noncomputable section
variable {Ω : Type*} [Fintype Ω]
def totalVariation (μ ν : FiniteDistribution Ω) : ℝ :=
(∑ x, |μ.weight x - ν.weight x|) / 2
theorem totalVariation_nonnegative (μ ν : FiniteDistribution Ω) :
0 ≤ μ.totalVariation ν :=
div_nonneg (Finset.sum_nonneg fun _ _ => abs_nonneg _) (by norm_num)
theorem totalVariation_comm (μ ν : FiniteDistribution Ω) :
μ.totalVariation ν = ν.totalVariation μ := by
unfold totalVariation
congr 1
apply Finset.sum_congr rfl
intro x _
exact abs_sub_comm _ _
theorem probability_sub_le_totalVariation (μ ν : FiniteDistribution Ω)
(event : Ω → Bool) :
μ.probability event - ν.probability event ≤ μ.totalVariation ν := by
have pointwise (x : Ω) :
2 * ((if event x then μ.weight x else 0) -
(if event x then ν.weight x else 0)) ≤
|μ.weight x - ν.weight x| + (μ.weight x - ν.weight x) := by
have h₁ := le_abs_self (μ.weight x - ν.weight x)
have h₂ := neg_le_abs (μ.weight x - ν.weight x)
cases event x <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> linarith
have summed := Finset.sum_le_sum (s := Finset.univ) fun x _ => pointwise x
simp only [← Finset.mul_sum, Finset.sum_add_distrib, Finset.sum_sub_distrib,
μ.normalized, ν.normalized, sub_self, add_zero] at summed
unfold probability totalVariation
linarith
theorem probability_abs_sub_le_totalVariation (μ ν : FiniteDistribution Ω)
(event : Ω → Bool) :
|μ.probability event - ν.probability event| ≤ μ.totalVariation ν := by
apply abs_le.mpr
constructor
· have h := ν.probability_sub_le_totalVariation μ event
rw [totalVariation_comm] at h
linarith
· exact μ.probability_sub_le_totalVariation ν event
theorem probability_le_add_totalVariation (μ ν : FiniteDistribution Ω)
(event : Ω → Bool) :
μ.probability event ≤ ν.probability event + μ.totalVariation ν := by
have h := μ.probability_sub_le_totalVariation ν event
linarith
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
end
end
end OAI
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