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import OAI.Computability.VertexCover.Information.GamesAdapter
import OAI.Computability.VertexCover.Information.Tensorization
namespace OAI
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namespace UniqueGames.Foundations.Repetition
open scoped BigOperators
open Games Information
noncomputable section
variable {Q₁ Q₂ A₁ A₂ : Type*}
[Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂]
[DecidableEq Q₁] [DecidableEq Q₂]
{n : Nat}
def selectedTupleLikelihood (G : Game Q₁ Q₂ A₁ A₂)
(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
(selected : Finset (Fin n)) (questions : Fin n → Q₁ × Q₂) : ℝ :=
if G.selectedWins strategy selected (Game.tupleQuestionEquiv n questions) then 1 else 0
omit [DecidableEq Q₁] [DecidableEq Q₂] in
theorem selectedTupleLikelihood_nonnegative (G : Game Q₁ Q₂ A₁ A₂)
(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
(selected : Finset (Fin n)) (questions : Fin n → Q₁ × Q₂) :
0 ≤ selectedTupleLikelihood G strategy selected questions := by
unfold selectedTupleLikelihood
split <;> norm_num
omit [DecidableEq Q₁] [DecidableEq Q₂] in
theorem selectedTupleLikelihood_le_one (G : Game Q₁ Q₂ A₁ A₂)
(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
(selected : Finset (Fin n)) (questions : Fin n → Q₁ × Q₂) :
selectedTupleLikelihood G strategy selected questions ≤ 1 := by
unfold selectedTupleLikelihood
split <;> norm_num
omit [DecidableEq Q₁] [DecidableEq Q₂] in
theorem selectedTupleLikelihood_mass (G : Game Q₁ Q₂ A₁ A₂)
(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
(selected : Finset (Fin n)) :
(∑ questions, independentProduct (fun _ : Fin n => G.questions.weight) questions *
selectedTupleLikelihood G strategy selected questions) =
G.selectedSuccess strategy selected := by
classical
change _ = ((G.questions.iid n).transport (Game.tupleQuestionEquiv n)).probability
(G.selectedWins strategy selected)
rw [FiniteDistribution.probability_transport]
simp only [FiniteDistribution.probability, FiniteDistribution.iid, independentProduct,
selectedTupleLikelihood, mul_ite, mul_one, mul_zero]
def selectedQuestionMarginal (G : Game Q₁ Q₂ A₁ A₂)
(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
(selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
(i : Fin n) : FiniteDistribution (Q₁ × Q₂) :=
((G.repetition n).questions.condition (G.selectedWins strategy selected) positive).pushforward
(fun questions => (questions.1 i, questions.2 i))
theorem selectedQuestionMarginal_weight (G : Game Q₁ Q₂ A₁ A₂)
(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
(selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected)
(i : Fin n) :
(selectedQuestionMarginal G strategy selected positive i).weight =
coordinateMarginal
(posterior (independentProduct (fun _ : Fin n => G.questions.weight))
(selectedTupleLikelihood G strategy selected) (G.selectedSuccess strategy selected)) i := by
classical
funext a
simp only [selectedQuestionMarginal, FiniteDistribution.pushforward,
FiniteDistribution.condition, coordinateMarginal]
refine Fintype.sum_equiv (Game.tupleQuestionEquiv (Q₁ := Q₁) (Q₂ := Q₂) n).symm _ _ ?_
intro questions
cases he : G.selectedWins strategy selected questions <;>
simp [Game.repetition_question_weight, Game.tupleQuestionEquiv, posterior, independentProduct, selectedTupleLikelihood, Game.selectedSuccess, he]
theorem selectedQuestionMarginal_totalVariation_sq_sum_le_log
(G : Game Q₁ Q₂ A₁ A₂)
(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
(selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected) :
(∑ i : Fin n,
(selectedQuestionMarginal G strategy selected positive i).totalVariation G.questions ^ 2) ≤
Real.log (1 / G.selectedSuccess strategy selected) := by
classical
change (∑ i : Fin n, Information.totalVariation
(selectedQuestionMarginal G strategy selected positive i).weight G.questions.weight ^ 2) ≤ _
simp_rw [selectedQuestionMarginal_weight]
exact posterior_coordinate_totalVariation_sq_sum_le_log
(fun _ : Fin n => G.questions.weight) (fun _ => gameLaw_isProbability G.questions)
(selectedTupleLikelihood G strategy selected)
(selectedTupleLikelihood_nonnegative G strategy selected)
(selectedTupleLikelihood_le_one G strategy selected)
positive (selectedTupleLikelihood_mass G strategy selected)
theorem unselectedQuestionMarginal_totalVariation_sq_sum_le_log
(G : Game Q₁ Q₂ A₁ A₂)
(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
(selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected) :
(∑ i : {i : Fin n // i ∉ selected},
(selectedQuestionMarginal G strategy selected positive i.1).totalVariation G.questions ^ 2) ≤
Real.log (1 / G.selectedSuccess strategy selected) := by
classical
let d : Fin n → ℝ := fun i =>
(selectedQuestionMarginal G strategy selected positive i).totalVariation G.questions
have hfull := selectedQuestionMarginal_totalVariation_sq_sum_le_log G strategy selected positive
have hsplit := Fintype.sum_subtype_add_sum_subtype (fun i : Fin n => i ∈ selected)
(fun i => d i ^ 2)
have hinst : Subtype.fintype (fun i : Fin n => i ∈ selected) =
Finset.Subtype.fintype selected := Subsingleton.elim _ _
rw [hinst] at hsplit
have hselected : 0 ≤ ∑ i : {i : Fin n // i ∈ selected}, d i.1 ^ 2 :=
Finset.sum_nonneg (fun _ _ => sq_nonneg _)
change (∑ i : {i : Fin n // i ∉ selected}, d i.1 ^ 2) ≤ _
change (∑ i : Fin n, d i ^ 2) ≤ _ at hfull
linarith
theorem unselectedQuestionMarginal_totalVariation_sum_le_sqrt
(G : Game Q₁ Q₂ A₁ A₂)
(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
(selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected) :
(∑ i : {i : Fin n // i ∉ selected},
(selectedQuestionMarginal G strategy selected positive i.1).totalVariation G.questions) ≤
Real.sqrt ((Fintype.card {i : Fin n // i ∉ selected} : ℝ) *
Real.log (1 / G.selectedSuccess strategy selected)) := by
classical
let d : {i : Fin n // i ∉ selected} → ℝ := fun i =>
(selectedQuestionMarginal G strategy selected positive i.1).totalVariation G.questions
have hsq := unselectedQuestionMarginal_totalVariation_sq_sum_le_log G strategy selected positive
have hcs := Finset.sum_mul_sq_le_sq_mul_sq
(Finset.univ : Finset {i : Fin n // i ∉ selected}) d (fun _ => (1 : ℝ))
simp only [mul_one, one_pow, Finset.sum_const, Finset.card_univ, nsmul_eq_mul, mul_one] at hcs
have hln : 0 ≤ Real.log (1 / G.selectedSuccess strategy selected) :=
(Finset.sum_nonneg (fun _ _ => sq_nonneg _)).trans hsq
have hcard : 0 ≤ (Fintype.card {i : Fin n // i ∉ selected} : ℝ) := Nat.cast_nonneg _
have hmul := mul_le_mul_of_nonneg_left hsq hcard
have hsqrt := Real.sq_sqrt (mul_nonneg hcard hln)
have hsqrtpos := Real.sqrt_nonneg
((Fintype.card {i : Fin n // i ∉ selected} : ℝ) *
Real.log (1 / G.selectedSuccess strategy selected))
change (∑ i, d i) ≤ _
change (∑ i, d i ^ 2) ≤ _ at hsq
change (Fintype.card {i : Fin n // i ∉ selected} : ℝ) * (∑ i, d i ^ 2) ≤ _ at hmul
nlinarith
theorem unselectedQuestionMarginal_totalVariation_sum_le_sqrt_sub_card
(G : Game Q₁ Q₂ A₁ A₂)
(strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
(selected : Finset (Fin n)) (positive : 0 < G.selectedSuccess strategy selected) :
(∑ i : {i : Fin n // i ∉ selected},
(selectedQuestionMarginal G strategy selected positive i.1).totalVariation G.questions) ≤
Real.sqrt (((n - selected.card : Nat) : ℝ) *
Real.log (1 / G.selectedSuccess strategy selected)) := by
classical
have hcard : Fintype.card {i : Fin n // i ∉ selected} = n - selected.card := by
simp
simpa only [hcard] using
unselectedQuestionMarginal_totalVariation_sum_le_sqrt G strategy selected positive
end
end UniqueGames.Foundations.Repetition
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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