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8df23f4 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | 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
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