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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 | import OAI.Combinatorics.BalancedRyser.Probability.FiniteUniformFibers
namespace OAI
namespace Balanced.AlignedListProbability
noncomputable section
attribute [local instance] Classical.propDecidable
theorem filter_outer_ratio {Ω : Type*} (p : PMF Ω) (C E : Set Ω)
(h : ∃ x ∈ C, x ∈ p.support) :
(p.filter C h).toOuterMeasure E =
p.toOuterMeasure (C ∩ E) / p.toOuterMeasure C := by
simp only [PMF.toOuterMeasure_apply, div_eq_mul_inv]
rw [← ENNReal.tsum_mul_right]
apply tsum_congr
intro x
by_cases hc : x ∈ C <;> by_cases he : x ∈ E <;>
simp [Set.indicator, hc, he]
theorem supported_real_ne_zero {Ω : Type*} (p : PMF Ω) (C : Set Ω)
(h : ∃ x ∈ C, x ∈ p.support) : (p.toOuterMeasure C).toReal ≠ 0 := by
apply ENNReal.toReal_ne_zero.mpr
constructor
· intro hz
have hd := (p.toOuterMeasure_apply_eq_zero_iff C).mp hz
obtain ⟨x, hx, hs⟩ := h
exact Set.disjoint_left.mp hd hs hx
· rw [PMF.toOuterMeasure_apply]
exact p.tsum_coe_indicator_ne_top C
theorem marginal_conditional_ratio {Ω Y : Type*} (p : PMF Ω) (q : PMF Y)
(C : Set Ω) (hC : ∃ x ∈ C, x ∈ p.support) (f : Ω → Y)
(hm : (p.filter C hC).map f = q) (B E : Set Y) :
(p.toOuterMeasure (C ∩ f ⁻¹' (B ∩ E))).toReal /
(p.toOuterMeasure (C ∩ f ⁻¹' B)).toReal =
(q.toOuterMeasure (B ∩ E)).toReal / (q.toOuterMeasure B).toReal := by
have ht (K : Set Y) :
(p.toOuterMeasure (C ∩ f ⁻¹' K)).toReal / (p.toOuterMeasure C).toReal =
(q.toOuterMeasure K).toReal := by
have h := congrArg (fun law => (law.toOuterMeasure K).toReal) hm
rw [PMF.toOuterMeasure_map_apply, filter_outer_ratio, ENNReal.toReal_div] at h
exact h
calc
_ = ((p.toOuterMeasure (C ∩ f ⁻¹' (B ∩ E))).toReal /
(p.toOuterMeasure C).toReal) /
((p.toOuterMeasure (C ∩ f ⁻¹' B)).toReal /
(p.toOuterMeasure C).toReal) :=
(div_div_div_cancel_right₀ (supported_real_ne_zero p C hC) _ _).symm
_ = _ := by rw [ht, ht]
end
end Balanced.AlignedListProbability
end OAI
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