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a2ee19a | 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 | import OAI.Probability.PosteriorReplicas.Information.ObservableLikelihood
namespace OAI
open MeasureTheory ProbabilityTheory InformationTheory Set Filter
open scoped ENNReal NNReal
noncomputable section
namespace PosteriorReplicas
lemma normalized_log_likelihood {X : Type*} [MeasurableSpace X] (ν : Measure X)
{g : X → ℝ≥0∞} (hg : Measurable g) (hpos : ∀ᵐ x ∂ν, 0 < g x)
(hmean : (∫⁻ x, g x ∂ν) = 1) :
Integrable (fun x => Real.exp (Real.log (g x).toReal)) ν ∧
(∫ x, Real.exp (Real.log (g x).toReal) ∂ν) = 1 := by
have ht : ∀ᵐ x ∂ν, g x < ∞ := ae_lt_top hg (by rw [hmean]; exact ENNReal.one_ne_top)
have he : (fun x => Real.exp (Real.log (g x).toReal)) =ᵐ[ν] (fun x => (g x).toReal) := by
filter_upwards [hpos,ht] with x hx hxt
exact Real.exp_log (ENNReal.toReal_pos hx.ne' hxt.ne)
refine ⟨(integrable_toReal_of_lintegral_ne_top hg.aemeasurable (by rw [hmean]; exact ENNReal.one_ne_top)).congr he.symm,?_⟩
rw [integral_congr_ae he,integral_toReal hg.aemeasurable ht,hmean,ENNReal.toReal_one]
lemma observableLikelihood_pos {d ℓ k m : ℕ} (hk : 1 ≤ k) (hkm : k+m ≤ ℓ)
{f : Vec d → ℝ≥0∞} {L : ℝ≥0∞} (hLt : L ≠ ∞) (hfL : ∀ x, f x ≤ L)
(z : Observation d ℓ k m) : 0 < observableLikelihood f z := by
unfold observableLikelihood
apply ENNReal.mul_pos_iff.mpr
constructor
· apply ENNReal.mul_pos_iff.mpr
constructor
· exact ENNReal.mul_pos_iff.mpr ⟨equalConstant_pos _ _, ENNReal.inv_pos.mpr (wishartMoment_ne_top hk hkm)⟩
· exact ENNReal.pow_pos (ENNReal.inv_pos.mpr ENNReal.ofReal_ne_top) _
· exact ENNReal.pow_pos (ENNReal.inv_pos.mpr (projectedDensity_ne_top .spherical hLt hfL _ _)) _
def observableLog {d ℓ k m : ℕ} (f : Vec d → ℝ≥0∞) (z : Observation d ℓ k m) : ℝ :=
Real.log (observableLikelihood f z).toReal
@[fun_prop] lemma measurable_observableLog {d ℓ k m : ℕ} {f : Vec d → ℝ≥0∞}
(hf : Measurable f) : Measurable (@observableLog d ℓ k m f) := by unfold observableLog; fun_prop
lemma observableLog_normalized {d ℓ k m : ℕ} (hd : k+m+2 ≤ d)
(hk : 1 ≤ k) (hm : 1 ≤ m) (hkm : k+m ≤ ℓ)
{f : Vec d → ℝ≥0∞} (hf : Measurable f) {L : ℝ≥0∞} (hLt : L ≠ ∞) (hfL : ∀ x, f x ≤ L)
[IsProbabilityMeasure ((sphereLaw d).withDensity f)] :
Integrable (fun z => Real.exp (observableLog f z))
(((equalReference (k := k) (m := m) ((sphereLaw d).withDensity f)).prod (auxLaw d ℓ k)).map observe) ∧
(∫ z, Real.exp (observableLog f z)
∂(((equalReference (k := k) (m := m) ((sphereLaw d).withDensity f)).prod (auxLaw d ℓ k)).map observe)) = 1 := by
apply normalized_log_likelihood _ (measurable_observableLikelihood hf)
(ae_of_all _ (observableLikelihood_pos hk hkm hLt hfL))
rw [lintegral_map (by fun_prop) measurable_observe]
exact observableLikelihood_lintegral hd hk hm hkm hf hLt hfL
end PosteriorReplicas
end
end OAI
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