| import Mathlib |
| import OAI.Computability.VertexCover.Repetition.SelectedJoint |
| import OAI.Computability.VertexCover.Repetition.PartialReveal |
|
|
| 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 selectedRawMarginal (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected}) : |
| (SelectedInput Q₁ Q₂ selected × SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) × |
| (Q₁ × Q₂) → ℝ := by |
| classical |
| exact fun z => ∑ u, if u j = z.2 then selectedJointWeight G strategy selected (z.1,u) else 0 |
|
|
| theorem selectedPosterior_eq_raw (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected}) (tv) (q) : |
| selectedSideWeight G strategy selected tv * coordinateMarginal |
| (posteriorOrOriginal |
| (independentProduct (fun i => (selectedInputProfile G selected tv.1 i).weight)) |
| (selectedLikelihood G strategy selected tv.1.1 tv.2) |
| (selectedSideMass G strategy selected tv)) j q = |
| selectedRawMarginal G strategy selected j (tv,q) := by |
| classical |
| rw [selectedSideWeight] |
| have h := weighted_coordinate_posterior_recombine |
| (independentProduct (fun i => (selectedInputProfile G selected tv.1 i).weight)) |
| (selectedLikelihood G strategy selected tv.1.1 tv.2) |
| (independentProduct_isProbability _ |
| (fun i => gameLaw_isProbability (selectedInputProfile G selected tv.1 i))) |
| (selectedLikelihood_nonnegative G strategy selected tv.1.1 tv.2) |
| («c» := selectedSideMass G strategy selected tv) rfl |
| ((selectedInputLaw G selected).weight tv.1) (G.selectedSuccess strategy selected) j q |
| rw [h] |
| simp only [Finset.mul_sum, selectedRawMarginal, selectedJointWeight] |
| apply Finset.sum_congr rfl |
| intro u _ |
| by_cases hu : u j = q |
| · simp only [ite_eq_left hu] |
| ring |
| · simp [hu] |
|
|
| theorem selectedRawMarginal_firstMarginal (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected}) (tv) : |
| firstMarginal (selectedRawMarginal G strategy selected j) tv = |
| selectedSideWeight G strategy selected tv := by |
| classical |
| simp only [firstMarginal, selectedRawMarginal] |
| rw [Finset.sum_comm] |
| simp only [Finset.sum_ite_eq, Finset.mem_univ, ite_true] |
| exact selectedJointWeight_firstMarginal G strategy selected tv |
|
|
| theorem selectedRawMarginal_isProbability (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) |
| (j : {i : Fin n // i ∉ selected}) : |
| IsProbability (selectedRawMarginal G strategy selected j) := by |
| classical |
| constructor |
| · intro z |
| apply Finset.sum_nonneg |
| intro u _ |
| split |
| · exact selectedJointWeight_nonnegative G strategy selected (z.1,u) |
| · exact le_rfl |
| · rw [Fintype.sum_prod_type] |
| change (∑ tv, firstMarginal (selectedRawMarginal G strategy selected j) tv) = 1 |
| simp_rw [selectedRawMarginal_firstMarginal, selectedSideWeight, div_eq_mul_inv] |
| rw [← Finset.sum_mul, selectedSideMass_total, mul_inv_cancel₀ positive.ne'] |
| |
| theorem selected_raw_information_bound [Nonempty A₁] [Nonempty A₂] |
| (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) : |
| (∑ j : {i : Fin n // i ∉ selected}, totalVariation |
| (selectedRawMarginal G strategy selected j) |
| (fun z => selectedSideWeight G strategy selected z.1 * |
| (selectedInputProfile G selected z.1.1 j).weight z.2)) ≤ |
| Real.sqrt ((Fintype.card {i : Fin n // i ∉ selected} : ℝ) * |
| (Real.log (Fintype.card (SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) : ℝ) + |
| Real.log (1 / G.selectedSuccess strategy selected))) := by |
| have h := selected_information_bound G strategy selected positive |
| simp_rw [selectedPosterior_eq_raw] at h |
| exact h |
| |
| def selectedOutsideLikelihood (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) |
| (t : (selected → Q₁ × Q₂) × SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) |
| (u : {i : Fin n // i ∉ selected} → Q₁ × Q₂) : ℝ := |
| (∏ i : selected, G.questions.weight (t.1 i)) * |
| selectedLikelihood G strategy selected t.1 t.2 u / G.selectedSuccess strategy selected |
| |
| def selectedObservationEquiv (selected : Finset (Fin n)) : |
| ((SelectedInput Q₁ Q₂ selected × SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) × |
| (Q₁ × Q₂)) ≃ |
| ((((selected → Q₁ × Q₂) × SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) × |
| ({i : Fin n // i ∉ selected} → Q₁ ⊕ Q₂)) × (Q₁ × Q₂)) where |
| toFun z := (((z.1.1.1,z.1.2),z.1.1.2),z.2) |
| invFun z := (((z.1.1.1,z.1.2),z.1.1.2),z.2) |
| left_inv _ := rfl |
| right_inv _ := rfl |
| |
| theorem selectedRawMarginal_fullReveal (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected}) (z) : |
| selectedRawMarginal G strategy selected j |
| ((selectedObservationEquiv (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected).symm z) = |
| fullRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected) z := by |
| classical |
| change selectedRawMarginal G strategy selected j |
| (((z.1.1.1,z.1.2),z.1.1.2),z.2) = _ |
| simp only [selectedRawMarginal, fullRevealMarginal] |
| apply Finset.sum_congr rfl |
| intro u _ |
| by_cases hu : u j = z.2 |
| · simp only [ite_eq_left hu, selectedJointWeight_factorization, selectedOutsideLikelihood] |
| ring |
| · simp [hu] |
| |
| theorem selectedRawReference_fullReveal (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected}) (z) : |
| (fun w => selectedSideWeight G strategy selected w.1 * |
| (selectedInputProfile G selected w.1.1 j).weight w.2) |
| ((selectedObservationEquiv (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected).symm z) = |
| fullRevealModel G.questions j (selectedOutsideLikelihood G strategy selected) z := by |
| change selectedSideWeight G strategy selected ((z.1.1.1,z.1.2),z.1.1.2) * |
| (revealProfile G.questions (z.1.2 j)).weight z.2 = _ |
| rw [← selectedRawMarginal_firstMarginal G strategy selected j] |
| unfold firstMarginal fullRevealModel |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro q _ |
| exact selectedRawMarginal_fullReveal G strategy selected j (z.1,q) |
| |
| theorem selected_raw_error_eq_fullReveal (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) (j : {i : Fin n // i ∉ selected}) : |
| totalVariation (selectedRawMarginal G strategy selected j) |
| (fun z => selectedSideWeight G strategy selected z.1 * |
| (selectedInputProfile G selected z.1.1 j).weight z.2) = |
| totalVariation (fullRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected)) |
| (fullRevealModel G.questions j (selectedOutsideLikelihood G strategy selected)) := by |
| rw [← totalVariation_comp_equiv |
| (selectedObservationEquiv (Q₁ := Q₁) (Q₂ := Q₂) (A₁ := A₁) (A₂ := A₂) selected).symm] |
| congr 1 <;> funext z |
| · exact selectedRawMarginal_fullReveal G strategy selected j z |
| · exact selectedRawReference_fullReveal G strategy selected j z |
| |
| theorem selected_fullReveal_information_bound [Nonempty A₁] [Nonempty A₂] |
| (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) : |
| (∑ j : {i : Fin n // i ∉ selected}, |
| totalVariation (fullRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected)) |
| (fullRevealModel G.questions j (selectedOutsideLikelihood G strategy selected))) ≤ |
| Real.sqrt ((Fintype.card {i : Fin n // i ∉ selected} : ℝ) * |
| (Real.log (Fintype.card (SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) : ℝ) + |
| Real.log (1 / G.selectedSuccess strategy selected))) := by |
| have h := selected_raw_information_bound G strategy selected positive |
| simp_rw [selected_raw_error_eq_fullReveal] at h |
| exact h |
| |
| def selectedInformationRadius (G : Game Q₁ Q₂ A₁ A₂) |
| (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂)) |
| (selected : Finset (Fin n)) : ℝ := |
| Real.sqrt ((Fintype.card {i : Fin n // i ∉ selected} : ℝ) * |
| (Real.log (Fintype.card (SelectedLabels (A₁ := A₁) (A₂ := A₂) selected) : ℝ) + |
| Real.log (1 / G.selectedSuccess strategy selected))) |
| |
| theorem selected_leftReveal_error_sum [Nonempty A₁] [Nonempty A₂] |
| (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) : |
| (∑ j : {i : Fin n // i ∉ selected}, |
| totalVariation (partialRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected)) |
| (leftRevealModel G.questions |
| (partialRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected)))) ≤ |
| 2 * selectedInformationRadius G strategy selected := by |
| calc |
| _ ≤ ∑ j : {i : Fin n // i ∉ selected}, |
| 2 * totalVariation (fullRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected)) |
| (fullRevealModel G.questions j (selectedOutsideLikelihood G strategy selected)) := by |
| apply Finset.sum_le_sum |
| intro j _ |
| exact leftReveal_full_error G.questions j _ |
| _ = 2 * ∑ j : {i : Fin n // i ∉ selected}, |
| totalVariation (fullRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected)) |
| (fullRevealModel G.questions j (selectedOutsideLikelihood G strategy selected)) := |
| (Finset.mul_sum _ _ _).symm |
| _ ≤ _ := mul_le_mul_of_nonneg_left |
| (selected_fullReveal_information_bound G strategy selected positive) (by norm_num) |
| |
| theorem selected_rightReveal_error_sum [Nonempty A₁] [Nonempty A₂] |
| (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) : |
| (∑ j : {i : Fin n // i ∉ selected}, |
| totalVariation (partialRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected)) |
| (rightRevealModel G.questions |
| (partialRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected)))) ≤ |
| 2 * selectedInformationRadius G strategy selected := by |
| calc |
| _ ≤ ∑ j : {i : Fin n // i ∉ selected}, |
| 2 * totalVariation (fullRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected)) |
| (fullRevealModel G.questions j (selectedOutsideLikelihood G strategy selected)) := by |
| apply Finset.sum_le_sum |
| intro j _ |
| exact rightReveal_full_error G.questions j _ |
| _ = 2 * ∑ j : {i : Fin n // i ∉ selected}, |
| totalVariation (fullRevealMarginal G.questions j (selectedOutsideLikelihood G strategy selected)) |
| (fullRevealModel G.questions j (selectedOutsideLikelihood G strategy selected)) := |
| (Finset.mul_sum _ _ _).symm |
| _ ≤ _ := mul_le_mul_of_nonneg_left |
| (selected_fullReveal_information_bound G strategy selected positive) (by norm_num) |
| |
| end |
| end UniqueGames.Foundations.Repetition |
| end |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
