| import Mathlib.Tactic.Linarith | |
| import Mathlib.Tactic.Ring | |
| import OAI.Computability.BinPacking.Information.GamesAdapter | |
| namespace OAI | |
| namespace BinPackingGames.Foundations.Repetition.ProfileCorrection | |
| open scoped BigOperators | |
| open BinPackingGames.Foundations.Information | |
| noncomputable section | |
| variable {S X Y : Type*} [Fintype S] [Fintype X] [Fintype Y] | |
| 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))) | |
| 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 | |
| omit [Fintype X] in | |
| theorem seedQuestionMarginal_first [Fintype X] (p : X × (S × Y) → ℝ) : | |
| firstMarginal (seedQuestionMarginal p) = firstMarginal p := by | |
| funext x | |
| simp only [seedQuestionMarginal, firstMarginal, Fintype.sum_prod_type] | |
| 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 | |
| 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 BinPackingGames.Foundations.Repetition.ProfileCorrection | |
| end OAI | |