openAI-math-dataset / lean /OAI /Probability /PosteriorReplicas /Information /NormalizedLikelihood.lean
| 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 | |