| import Mathlib.Probability.ProbabilityMassFunction.Integrals |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet.FiniteLaw |
|
|
| open MeasureTheory |
| open scoped BigOperators |
| |
| noncomputable def weightPMF {α : Type*} [Fintype α] (p : α → ℝ) |
| (hp : ∀ a, 0 ≤ p a) (hp1 : ∑ a, p a = 1) : PMF α := |
| PMF.ofFintype (fun a => ENNReal.ofReal (p a)) (by |
| rw [← ENNReal.ofReal_sum_of_nonneg (fun a _ => hp a), hp1] |
| exact ENNReal.ofReal_one) |
|
|
| noncomputable def weightMeasure {α : Type*} [Fintype α] [MeasurableSpace α] |
| (p : α → ℝ) (hp : ∀ a, 0 ≤ p a) (hp1 : ∑ a, p a = 1) : Measure α := |
| (weightPMF p hp hp1).toMeasure |
|
|
| noncomputable def kernelJointPMF {α β : Type*} [Fintype α] [Fintype β] |
| (m : β → ℝ) (k : β → α → ℝ) |
| (hm : ∀ b, 0 ≤ m b) (hm1 : ∑ b, m b = 1) |
| (hk : ∀ b a, 0 ≤ k b a) (hk1 : ∀ b, ∑ a, k b a = 1) : PMF (β × α) := |
| weightPMF (fun ba => m ba.1 * k ba.1 ba.2) |
| (fun ba => mul_nonneg (hm ba.1) (hk ba.1 ba.2)) (by |
| rw [Fintype.sum_prod_type] |
| simp_rw [← Finset.mul_sum, hk1, mul_one] |
| exact hm1) |
|
|
| noncomputable def kernelJointMeasure {α β : Type*} [Fintype α] [Fintype β] |
| [MeasurableSpace α] [MeasurableSpace β] |
| (m : β → ℝ) (k : β → α → ℝ) |
| (hm : ∀ b, 0 ≤ m b) (hm1 : ∑ b, m b = 1) |
| (hk : ∀ b a, 0 ≤ k b a) (hk1 : ∀ b, ∑ a, k b a = 1) : Measure (β × α) := |
| (kernelJointPMF m k hm hm1 hk hk1).toMeasure |
|
|
| end MatroidProphet.FiniteLaw |
|
|
| end OAI |
|
|
