| import Mathlib |
| import OAI.Combinatorics.IndependentSets.Repetition.TotalVariation |
| import OAI.Combinatorics.IndependentSets.Repetition.Pinsker |
|
|
| namespace OAI |
|
|
| namespace IndependentSetsGames.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 IndependentSetsGames.Foundations.Repetition |
|
|
| end OAI |
|
|
