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f0f05de | 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 | import OAI.Combinatorics.BalancedRyser.Probability.ConditionalOrderVariance
namespace OAI
open scoped BigOperators ENNReal
namespace Balanced.ConditionalOrderLaw
theorem filter_uniform_eq {Ω : Type*} [Fintype Ω] [Nonempty Ω]
(s : Finset Ω) (hs : s.Nonempty)
(h : ∃ a ∈ (s : Set Ω), a ∈ (PMF.uniformOfFintype Ω).support) :
(PMF.uniformOfFintype Ω).filter (s : Set Ω) h = PMF.uniformOfFinset s hs := by
classical
have hn : (Fintype.card Ω : ℝ≥0∞) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero
have hs0 : (s.card : ℝ≥0∞) ≠ 0 := by exact_mod_cast hs.card_pos.ne'
have hsum : (∑' a : Ω, (s : Set Ω).indicator (PMF.uniformOfFintype Ω) a) =
(s.card : ℝ≥0∞) * (Fintype.card Ω : ℝ≥0∞)⁻¹ := by
simp [tsum_fintype, Set.indicator, Finset.sum_ite_mem]
ext a
rw [PMF.filter_apply, hsum, PMF.uniformOfFinset_apply]
by_cases ha : a ∈ s
· simp only [Set.indicator_of_mem ha, PMF.uniformOfFintype_apply, ite_eq_left ha]
rw [ENNReal.mul_inv (Or.inl hs0) (Or.inl (ENNReal.natCast_ne_top _)), inv_inv]
calc
_ = (s.card : ℝ≥0∞)⁻¹ *
((Fintype.card Ω : ℝ≥0∞)⁻¹ * Fintype.card Ω) := by ac_rfl
_ = _ := by rw [ENNReal.inv_mul_cancel hn (ENNReal.natCast_ne_top _), mul_one]
· simp [ha]
open ConditionalOrderVariance MeasureTheory
variable {F : Type*} [Field F] [Fintype F] [DecidableEq F]
omit [Field F] [Fintype F] in
theorem target_event_supported (U V : Finset (Line F)) (i x : U) :
∃ p ∈ {p : OrderPair U V | p.1 i = x},
p ∈ (PMF.uniformOfFintype (OrderPair U V)).support := by
exact ⟨(Equiv.swap i x, Equiv.refl V), by simp, by simp⟩
omit [Field F] [Fintype F] in
theorem conditioned_eq_filter (U V : Finset (Line F)) (i x : U) :
conditionedOrderLaw U V i x =
(PMF.uniformOfFintype (OrderPair U V)).filter
{p | p.1 i = x} (target_event_supported U V i x) := by
classical
let s := (Finset.univ.filter (fun σ : Equiv.Perm U => σ i = x)) ×ˢ
(Finset.univ : Finset (Equiv.Perm V))
have hs : s.Nonempty := ⟨(Equiv.swap i x, Equiv.refl V), by simp [s]⟩
have hset : (s : Set (OrderPair U V)) = {p | p.1 i = x} := by
ext p
simp [s]
have hsupport : ∃ p ∈ (s : Set (OrderPair U V)),
p ∈ (PMF.uniformOfFintype (OrderPair U V)).support := by
rw [hset]
exact target_event_supported U V i x
change PMF.uniformOfFinset s hs = _
have h := (filter_uniform_eq s hs hsupport).symm
simpa only [hset] using h
omit [Field F] in
theorem target_event_probability (U V : Finset (Line F))
[MeasurableSpace (OrderPair U V)] [MeasurableSingletonClass (OrderPair U V)]
(i x : U) :
(PMF.uniformOfFintype (OrderPair U V)).toMeasure.real {p | p.1 i = x} =
1 / (U.card : ℝ) := by
classical
rw [← integral_indicator_one (Set.to_countable _).measurableSet,
PMF.uniformOfFintype, WithoutReplacement.integral_uniformOfFinset,
← Finset.univ_product_univ, Finset.expect_product]
simp only [Set.indicator, Set.mem_ofPred_eq, Pi.one_apply, Fintype.expect_const]
simpa only [Fintype.card_coe] using PermutationSubsetLaw.target_probability i x
omit [Field F] in
theorem target_event_probability_pos (U V : Finset (Line F))
[MeasurableSpace (OrderPair U V)] [MeasurableSingletonClass (OrderPair U V)]
(i x : U) :
0 < (PMF.uniformOfFintype (OrderPair U V)).toMeasure.real {p | p.1 i = x} := by
rw [target_event_probability]
have hU : (0 : ℝ) < U.card := by
exact_mod_cast (Finset.card_pos.mpr ⟨i.val, i.property⟩)
positivity
omit [Field F] [Fintype F] [DecidableEq F] in
theorem uniform_pair_atom {α β : Type*} [Fintype α] [Nonempty α]
[Fintype β] [Nonempty β] (a : α) (b : β) :
PMF.uniformOfFintype (α × β) (a, b) =
PMF.uniformOfFintype α a * PMF.uniformOfFintype β b := by
simp only [PMF.uniformOfFintype_apply, Fintype.card_prod, Nat.cast_mul]
exact ENNReal.mul_inv (Or.inl (by exact_mod_cast Fintype.card_ne_zero))
(Or.inl (ENNReal.natCast_ne_top _))
end Balanced.ConditionalOrderLaw
end OAI
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