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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 | import OAI.Probability.FreeIsing.Conditioning.FiniteBayes
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped ENNReal ProbabilityTheory
namespace Problem367.FiniteBayes
theorem posterior_integral
{A X : Type*} [MeasurableSpace A] [MeasurableSpace X]
[Fintype A] [Nonempty A] [StandardBorelSpace A]
[MeasurableSingletonClass A]
(mu : Measure A) [IsProbabilityMeasure mu]
(kappa : Kernel A X) [IsFiniteKernel kappa]
(nu : Measure X) [SigmaFinite nu]
(L : A → X → ℝ) (hL : ∀ a, Measurable (L a))
(hLpos : ∀ a x, 0 < L a x)
(hkappa : ∀ a, kappa a = nu.withDensity (fun x => ENNReal.ofReal (L a x)))
(q : A → ℝ) :
∀ᵐ x ∂kappa ∘ₘ mu,
(∫ a, q a ∂(kappa†mu) x) =
(∑ a, mu.real {a} * L a x * q a) / (∑ a, mu.real {a} * L a x) := by
have hpost := finite_posterior_singletons_of_positive mu nu kappa
(fun a x => ENNReal.ofReal (L a x))
(fun a => (hL a).ennreal_ofReal) hkappa
(fun a x => ne_of_gt (ENNReal.ofReal_pos.mpr (hLpos a x)))
(fun a x => ENNReal.ofReal_ne_top)
filter_upwards [hpost] with x hx
have hden : (∑ a, mu {a} * ENNReal.ofReal (L a x)).toReal =
∑ a, mu.real {a} * L a x := by
rw [ENNReal.toReal_sum (fun a _ =>
ENNReal.mul_ne_top (measure_ne_top mu {a}) ENNReal.ofReal_ne_top)]
apply Finset.sum_congr rfl
intro a _
rw [ENNReal.toReal_mul, ENNReal.toReal_ofReal (hLpos a x).le]
rfl
rw [integral_fintype Integrable.of_finite, Finset.sum_div]
apply Finset.sum_congr rfl
intro a _
change ((kappa†mu) x {a}).toReal * q a = _
rw [hx a, ENNReal.toReal_div, ENNReal.toReal_mul,
ENNReal.toReal_ofReal (hLpos a x).le, hden]
unfold Measure.real
ring
theorem posterior_integral_exp
{A X : Type*} [MeasurableSpace A] [MeasurableSpace X]
[Fintype A] [Nonempty A] [StandardBorelSpace A]
[MeasurableSingletonClass A]
(mu : Measure A) [IsProbabilityMeasure mu]
(kappa : Kernel A X) [IsFiniteKernel kappa]
(nu : Measure X) [SigmaFinite nu]
(E : A → X → ℝ) (hE : ∀ a, Measurable (E a))
(c : X → ℝ) (hc : Measurable c) (hcpos : ∀ x, 0 < c x)
(hkappa : ∀ a, kappa a =
nu.withDensity (fun x => ENNReal.ofReal (c x * Real.exp (E a x))))
(q : A → ℝ) :
∀ᵐ x ∂kappa ∘ₘ mu,
(∫ a, q a ∂(kappa†mu) x) =
(∑ a, mu.real {a} * Real.exp (E a x) * q a) /
(∑ a, mu.real {a} * Real.exp (E a x)) := by
have hpost := posterior_integral mu kappa nu
(fun a x => c x * Real.exp (E a x))
(fun a => hc.mul (hE a).exp)
(fun a x => mul_pos (hcpos x) (Real.exp_pos _)) hkappa q
filter_upwards [hpost] with x hx
rw [hx]
have hnum : (∑ a, mu.real {a} * (c x * Real.exp (E a x)) * q a) =
c x * ∑ a, mu.real {a} * Real.exp (E a x) * q a := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro a _
ring
have hden : (∑ a, mu.real {a} * (c x * Real.exp (E a x))) =
c x * ∑ a, mu.real {a} * Real.exp (E a x) := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro a _
ring
rw [hnum, hden, mul_div_mul_left _ _ (ne_of_gt (hcpos x))]
end Problem367.FiniteBayes
end
end OAI
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