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e63eb7f | 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 90 91 92 93 94 95 96 97 98 99 100 | import OAI.Combinatorics.EditApproximation.Refinement.PhysicalParentDraw
namespace OAI
/-! Exact coordinate-refresh identities for finite product probability laws. -/
namespace EditApproximation
open Finset MeasureTheory ProbabilityTheory
section FiniteProduct
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
{X : ι → Type*} [∀ i, Fintype (X i)] [∀ i, MeasurableSpace (X i)]
[∀ i, MeasurableSingletonClass (X i)]
(μ : ∀ i, Measure (X i)) [∀ i, IsProbabilityMeasure (μ i)]
/-- Refreshing one coordinate with its own independent law preserves the
complete finite product law. -/
theorem finite_pi_update_map (i : ι) :
((Measure.pi μ).prod (μ i)).map
(fun pair : (∀ j, X j) × X i => Function.update pair.1 i pair.2) = Measure.pi μ := by
apply Eq.symm
apply Measure.pi_eq
intro sets hsets
have hpreimage : (fun pair : (∀ j, X j) × X i => Function.update pair.1 i pair.2) ⁻¹'
Set.univ.pi sets = (Set.univ.pi (Function.update sets i Set.univ)) ×ˢ sets i := by
ext pair
constructor
· intro h
constructor
· intro j _
by_cases hji : j = i
· subst j
simp only [Function.update_self, Set.mem_univ]
· simpa only [Function.update_of_ne hji] using h j (Set.mem_univ j)
· simpa only [Function.update_self] using h i (Set.mem_univ i)
· rintro ⟨hold, hfresh⟩ j _
by_cases hji : j = i
· subst j
simpa only [Function.update_self] using hfresh
· simpa only [Function.update_of_ne hji] using hold j (Set.mem_univ j)
rw [Measure.map_apply (measurable_of_finite _) (MeasurableSet.univ_pi hsets),
hpreimage, Measure.prod_prod, Measure.pi_pi]
rw [← prod_erase_mul (s := univ) (fun j => μ j (Function.update sets i Set.univ j)) (mem_univ i),
← prod_erase_mul (s := univ) (fun j => μ j (sets j)) (mem_univ i)]
simp only [Function.update_self, measure_univ, mul_one]
congr 1
apply prod_congr rfl
intro j hj
simp only [Function.update_of_ne (ne_of_mem_erase hj)]
/-- Integrating a fresh coordinate, then the complete original array, is
exactly integration in the original product law. -/
theorem finite_pi_integral_update (i : ι) (payoff : (∀ j, X j) → ℝ) :
(∫ original, ∫ fresh, payoff (Function.update original i fresh) ∂μ i ∂Measure.pi μ) =
∫ original, payoff original ∂Measure.pi μ := by
rw [← integral_prod (fun pair : (∀ j, X j) × X i =>
payoff (Function.update pair.1 i pair.2)) (.of_finite)]
have hmap := finite_pi_update_map μ i
have hintegral := integral_map_of_stronglyMeasurable
(μ := (Measure.pi μ).prod (μ i))
(measurable_of_finite (fun pair : (∀ j, X j) × X i => Function.update pair.1 i pair.2))
(measurable_of_finite payoff).stronglyMeasurable
rw [hmap] at hintegral
exact hintegral.symm
/-- Two independent refreshed coordinates provide the local parent law.
The original arrays may still contain their previous values at those keys. -/
theorem finite_pi_integral_update_pair (i j : ι) (payoff : (∀ k, X k) → ℝ) :
(∫ original, ∫ fresh : X i × X j,
payoff (Function.update (Function.update original i fresh.1) j fresh.2)
∂(μ i).prod (μ j) ∂Measure.pi μ) =
∫ original, payoff original ∂Measure.pi μ := by
have hinner original : (∫ fresh : X i × X j,
payoff (Function.update (Function.update original i fresh.1) j fresh.2) ∂(μ i).prod (μ j)) =
∫ first, ∫ second, payoff (Function.update (Function.update original i first) j second) ∂μ j ∂μ i :=
integral_prod _ (.of_finite)
simp_rw [hinner]
rw [finite_pi_integral_update μ i
(fun original => ∫ second, payoff (Function.update original j second) ∂μ j)]
exact finite_pi_integral_update μ j payoff
/-- A pointwise bound after exact coordinate refresh integrates to the
unconditional bound in the original global product. -/
theorem finite_pi_integral_bound_of_update_pair (i j : ι)
(payoff budget : (∀ k, X k) → ℝ)
(hbound : ∀ original, (∫ fresh : X i × X j,
payoff (Function.update (Function.update original i fresh.1) j fresh.2) ∂(μ i).prod (μ j)) ≤
budget original) :
(∫ original, payoff original ∂Measure.pi μ) ≤ ∫ original, budget original ∂Measure.pi μ := by
rw [← finite_pi_integral_update_pair μ i j payoff]
exact integral_mono (.of_finite) (.of_finite) hbound
end FiniteProduct
end EditApproximation
end OAI
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