| import Mathlib |
| import OAI.Computability.VertexCover.Information.Conditionals |
| import OAI.Computability.VertexCover.Information.Kernels |
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| namespace OAI |
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| namespace UniqueGames.Foundations.Repetition.ProfileCorrection |
| open scoped BigOperators |
| open UniqueGames.Foundations.Information |
| noncomputable section |
| variable {S X Y : Type*} [Fintype S] [Fintype X] [Fintype Y] |
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| def seedQuestionMarginal (p : X × (S × Y) → ℝ) : X × S → ℝ := |
| firstMarginal (fun z : (X × S) × Y => p (z.1.1, (z.1.2, z.2))) |
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| theorem seedQuestionMarginal_isProbability (p : X × (S × Y) → ℝ) |
| (hp : IsProbability p) : IsProbability (seedQuestionMarginal p) := by |
| apply firstMarginal_isProbability |
| constructor |
| · intro z |
| exact hp.1 _ |
| · have h := hp.2 |
| simp only [Fintype.sum_prod_type] at h ⊢ |
| exact h |
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|
| omit [Fintype X] in |
| theorem seedQuestionMarginal_first (p : X × (S × Y) → ℝ) : |
| firstMarginal (seedQuestionMarginal p) = firstMarginal p := by |
| funext x |
| simp only [seedQuestionMarginal, firstMarginal, Fintype.sum_prod_type] |
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| theorem totalVariation_triangle {Ω : Type*} [Fintype Ω] |
| (p «c» d : Ω → ℝ) : |
| totalVariation p d ≤ totalVariation p «c» + totalVariation «c» d := by |
| unfold totalVariation |
| have h := Finset.sum_le_sum (s := Finset.univ) |
| (fun x _ => abs_sub_le (p x) («c» x) (d x)) |
| rw [Finset.sum_add_distrib] at h |
| linarith |
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| theorem left_profile_correction |
| (p : X × (S × Y) → ℝ) (μ : X × Y → ℝ) |
| (fallbackS : S → ℝ) (fallbackY : Y → ℝ) |
| (hp : IsProbability p) (hμ : IsProbability μ) |
| (hS : IsProbability fallbackS) (hY : IsProbability fallbackY) : |
| totalVariation p (fun z => μ (z.1, z.2.2) * |
| conditionalKernel (seedQuestionMarginal p) fallbackS z.1 z.2.1) ≤ |
| totalVariation p (fun z => seedQuestionMarginal p (z.1, z.2.1) * |
| conditionalKernel μ fallbackY z.1 z.2.2) + |
| totalVariation (firstMarginal p) (firstMarginal μ) := by |
| let σ := seedQuestionMarginal p |
| let L := conditionalKernel σ fallbackS |
| let M := conditionalKernel μ fallbackY |
| let k : X → S × Y → ℝ := fun x sy => L x sy.1 * M x sy.2 |
| let «c» : X × (S × Y) → ℝ := fun z => σ (z.1,z.2.1) * M z.1 z.2.2 |
| let d : X × (S × Y) → ℝ := fun z => μ (z.1,z.2.2) * L z.1 z.2.1 |
| have hσ : IsProbability σ := seedQuestionMarginal_isProbability p hp |
| have hk : ∀ x, IsProbability (k x) := by |
| intro x |
| exact kernelProduct_isProbability (L x) (fun _ : S => M x) |
| (conditionalKernel_isProbability σ fallbackS hσ hS x) |
| (fun _ => conditionalKernel_isProbability μ fallbackY hμ hY x) |
| have hc : «c» = fun z => firstMarginal σ z.1 * k z.1 z.2 := by |
| funext z |
| change σ (z.1,z.2.1) * M z.1 z.2.2 = |
| firstMarginal σ z.1 * (L z.1 z.2.1 * M z.1 z.2.2) |
| rw [← mul_assoc, marginal_mul_conditionalKernel σ fallbackS hσ] |
| have hd : d = fun z => firstMarginal μ z.1 * k z.1 z.2 := by |
| funext z |
| change μ (z.1,z.2.2) * L z.1 z.2.1 = |
| firstMarginal μ z.1 * (L z.1 z.2.1 * M z.1 z.2.2) |
| calc |
| _ = (firstMarginal μ z.1 * M z.1 z.2.2) * L z.1 z.2.1 := by |
| rw [marginal_mul_conditionalKernel μ fallbackY hμ] |
| _ = _ := by ring |
| have hmiddle : totalVariation «c» d = |
| totalVariation (firstMarginal σ) (firstMarginal μ) := by |
| rw [hc, hd] |
| exact totalVariation_joint_common_kernel _ _ k hk |
| have result := totalVariation_triangle p «c» d |
| rw [hmiddle] at result |
| have hmarg : firstMarginal σ = firstMarginal p := seedQuestionMarginal_first p |
| rw [hmarg] at result |
| exact result |
| end |
| end UniqueGames.Foundations.Repetition.ProfileCorrection |
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| end OAI |
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