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f61e547 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal ProbabilityTheory
namespace Problem367
theorem finite_observation_law_density
{A X : Type*} [Fintype A] [MeasurableSpace A] [MeasurableSingletonClass A]
[MeasurableSpace X] (π : Measure A) (ν : Measure X) (κ : Kernel A X)
(L : A → X → ℝ≥0∞) (hL : ∀ a, Measurable (L a))
(hκ : ∀ a, κ a = ν.withDensity (L a)) :
κ ∘ₘ π = ν.withDensity (fun x => ∑ a, π {a} * L a x) := by
ext s hs
rw [Measure.bind_apply hs κ.aemeasurable, withDensity_apply _ hs,
lintegral_fintype]
simp_rw [hκ, withDensity_apply _ hs]
rw [lintegral_finsetSum]
· apply Finset.sum_congr rfl
intro a ha
rw [lintegral_const_mul _ (hL a), mul_comm]
· intro a ha
exact measurable_const.mul (hL a)
theorem finite_posterior_singletons
{A X : Type*} [Fintype A] [MeasurableSpace A] [MeasurableSingletonClass A]
[StandardBorelSpace A] [Nonempty A] [MeasurableSpace X]
(π : Measure A) [IsFiniteMeasure π] (ν : Measure X) [SigmaFinite ν]
(κ : Kernel A X) [IsFiniteKernel κ]
(L : A → X → ℝ≥0∞) (hL : ∀ a, Measurable (L a))
(hκ : ∀ a, κ a = ν.withDensity (L a))
(hpos : ∀ᵐ x ∂ν, (∑ a, π {a} * L a x) ≠ 0)
(hfinite : ∀ᵐ x ∂ν, (∑ a, π {a} * L a x) ≠ ∞) :
∀ᵐ x ∂κ ∘ₘ π, ∀ a,
(posterior κ π) x {a} = π {a} * L a x / (∑ b, π {b} * L b x) := by
let Z : X → ℝ≥0∞ := fun x => ∑ a, π {a} * L a x
have hZ : Measurable Z := by
exact Finset.measurable_sum _ (fun a _ => measurable_const.mul (hL a))
have hdata : κ ∘ₘ π = ν.withDensity Z := finite_observation_law_density π ν κ L hL hκ
have hRNν (a : A) :
(κ a).rnDeriv (κ ∘ₘ π) =ᵐ[ν] fun x => L a x / Z x := by
rw [hdata]
have hright := Measure.rnDeriv_withDensity_right (κ a) ν hZ.aemeasurable hpos hfinite
have hleft : (κ a).rnDeriv ν =ᵐ[ν] L a := by
rw [hκ a]
exact Measure.rnDeriv_withDensity ν (hL a)
filter_upwards [hright, hleft] with x hr hl
rw [hr, hl, div_eq_mul_inv, mul_comm]
have hRN (a : A) :
(κ a).rnDeriv (κ ∘ₘ π) =ᵐ[κ ∘ₘ π] fun x => L a x / Z x := by
have hac : κ ∘ₘ π ≪ ν := by
rw [hdata]
exact withDensity_absolutelyContinuous ν Z
exact hac.ae_eq (hRNν a)
have hpost := posterior_eq_withDensity_of_countable κ π
filter_upwards [hpost, ae_all_iff.mpr hRN] with x hx hRNx
intro a
rw [hx]
simp only [withDensity_apply _ (measurableSet_singleton a), lintegral_singleton]
rw [hRNx a]
dsimp only [Z]
simp only [div_eq_mul_inv]
ac_rfl
theorem finite_posterior_singletons_of_positive
{A X : Type*} [Fintype A] [MeasurableSpace A] [MeasurableSingletonClass A]
[StandardBorelSpace A] [Nonempty A] [MeasurableSpace X]
(π : Measure A) [IsProbabilityMeasure π] (ν : Measure X) [SigmaFinite ν]
(κ : Kernel A X) [IsFiniteKernel κ]
(L : A → X → ℝ≥0∞) (hL : ∀ a, Measurable (L a))
(hκ : ∀ a, κ a = ν.withDensity (L a))
(hpos : ∀ a x, L a x ≠ 0) (hfinite : ∀ a x, L a x ≠ ∞) :
∀ᵐ x ∂κ ∘ₘ π, ∀ a,
(posterior κ π) x {a} = π {a} * L a x / (∑ b, π {b} * L b x) := by
have hp : ∃ a, π {a} ≠ 0 := by
by_contra! h
have hs : (∑ a, π {a}) = 0 := Finset.sum_eq_zero (fun a _ => h a)
simp at hs
obtain ⟨a, ha⟩ := hp
apply finite_posterior_singletons π ν κ L hL hκ
· apply ae_of_all
intro x
apply ne_of_gt
apply lt_of_lt_of_le (pos_iff_ne_zero.mpr (mul_ne_zero ha (hpos a x)))
exact Finset.single_le_sum (f := fun b => π {b} * L b x)
(fun b _ => zero_le) (Finset.mem_univ a)
· apply ae_of_all
intro x
exact ENNReal.sum_ne_top.mpr (fun b _ => ENNReal.mul_ne_top (measure_ne_top π _) (hfinite b x))
theorem finite_posterior_mean
{A X : Type*} [Fintype A] [MeasurableSpace A] [MeasurableSingletonClass A]
[StandardBorelSpace A] [Nonempty A] [MeasurableSpace X]
(π : Measure A) [IsProbabilityMeasure π] (ν : Measure X) [SigmaFinite ν]
(κ : Kernel A X) [IsFiniteKernel κ]
(L : A → X → ℝ≥0∞) (hL : ∀ a, Measurable (L a))
(hκ : ∀ a, κ a = ν.withDensity (L a))
(hpos : ∀ a x, L a x ≠ 0) (hfinite : ∀ a x, L a x ≠ ∞)
(f : A → ℝ) :
∀ᵐ x ∂κ ∘ₘ π,
(∫ a, f a ∂(posterior κ π) x) =
∑ a, f a * (π {a} * L a x / (∑ b, π {b} * L b x)).toReal := by
filter_upwards [finite_posterior_singletons_of_positive π ν κ L hL hκ hpos hfinite]
with x hx
rw [integral_fintype Integrable.of_finite]
apply Finset.sum_congr rfl
intro a ha
rw [measureReal_def, hx a, smul_eq_mul, mul_comm]
end Problem367
end
end OAI
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