| import Mathlib |
| import OAI.Probability.GaussianInformation.Kernel.MessageComponents |
|
|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Metric Set |
| open scoped ENNReal |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| open MeasureTheory ProbabilityTheory |
| open MeasureTheory |
| open scoped ENNReal |
|
|
| namespace KernelPotential.Noise |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| open Projection Projection.Support GaussianTools ReferenceGaussian DriftTools.FiniteReference Bayes |
|
|
| lemma likelihood_sum {α β ι : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| [Fintype ι] [MeasurableSpace ι] [MeasurableSingletonClass ι] |
| (Λ : Measure β) [SigmaFinite Λ] (K : β → α → ℝ≥0∞) (κ : β → ι → ℝ≥0∞) |
| (hK : Measurable (Function.uncurry K)) (hκ : Measurable (Function.uncurry κ)) |
| (hK1 : ∀ s,∫⁻ z,K z s ∂Λ=1) (hκ1 : ∀ z,∑ w,κ z w=1) (s : α) : |
| ∑ w,likelihood Λ K κ s w=1 := by |
| have hb (z : β) (w : ι) : κ z w ≤ 1 := by |
| rw [← hκ1 z]; exact Finset.single_le_sum (fun _ _ => bot_le) (Finset.mem_univ w) |
| have hfin (w : ι) : (∫⁻ z,K z s*κ z w ∂Λ) ≠ ∞ := |
| ne_of_lt (((likelihood_bounds Λ K κ hK hK1 (ε := 0) (fun _ _ => bot_le) hb s w).2).trans_lt (by norm_num)) |
| unfold likelihood |
| rw [← ENNReal.toReal_sum (fun w _ => hfin w),← lintegral_finsetSum] |
| · simp only [← Finset.mul_sum,hκ1,mul_one,hK1,ENNReal.toReal_one] |
| · intro w _; fun_prop |
|
|
| lemma likelihood_real_bounds {α β ι : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| [Fintype ι] [MeasurableSpace ι] [MeasurableSingletonClass ι] |
| (Λ : Measure β) [SigmaFinite Λ] (K : β → α → ℝ≥0∞) (κ : β → ι → ℝ≥0∞) |
| (hK : Measurable (Function.uncurry K)) (hK1 : ∀ s,∫⁻ z,K z s ∂Λ=1) |
| {ε : ℝ} (hε : 0 ≤ ε) (hκε : ∀ z w,ENNReal.ofReal ε ≤ κ z w) (hκ1 : ∀ z w,κ z w ≤ 1) |
| (s : α) (w : ι) : ε ≤ likelihood Λ K κ s w ∧ likelihood Λ K κ s w ≤ 1 := by |
| obtain ⟨hl,hu⟩ := likelihood_bounds Λ K κ hK hK1 hκε hκ1 s w |
| have hfin := ne_of_lt (hu.trans_lt (by norm_num : (1:ℝ≥0∞)<∞)) |
| exact ⟨by simpa [likelihood,ENNReal.toReal_ofReal hε] using ENNReal.toReal_mono hfin hl, |
| by simpa [likelihood] using ENNReal.toReal_mono ENNReal.one_ne_top hu⟩ |
|
|
| lemma likelihood_first {d k : ℕ} {η : ℝ} (hη : 0 < η) |
| (φ : ((Mat k d) × (Fin k → ℝ)) → ℝ) (hφ : Measurable φ) |
| (hφ0 : ∀ t,0 ≤ φ t) (hφ1 : ∀ t,φ t ≤ 1) (s : Ambient d) : |
| ((∫⁻ z : Observation k d,observationKernel η z s*ENNReal.ofReal (φ (z.1.1,z.2.1)) |
| ∂observationBase k d).toReal)= |
| ∫ X,∫ e,φ (X,projectVector X s+e) ∂noiseLaw k η ∂gaussianRows k d := by |
| have hm : Measurable (fun z : Observation k d => φ (z.1.1,z.2.1)) := |
| hφ.comp (measurable_fst.fst.prodMk measurable_snd.fst) |
| have hK := observationKernel_measurable (k := k) (d := d) η |
| have hKs : Measurable (fun z : Observation k d => observationKernel η z s) := by fun_prop |
| have he := first_rule_integral hη φ hφ hφ0 hφ1 s |
| rw [integral_withDensity_eq_integral_toReal_smul hKs |
| (ae_of_all _ (fun z => lt_top_iff_ne_top.mpr (observationKernel_ne_top hη z s)))] at he |
| have hi : Measurable (fun z : Observation k d => observationKernel η z s*ENNReal.ofReal (φ (z.1.1,z.2.1))) := by fun_prop |
| rw [← integral_toReal hi.aemeasurable (ae_of_all _ (fun z => ENNReal.mul_lt_top |
| (lt_top_iff_ne_top.mpr (observationKernel_ne_top hη z s)) ENNReal.ofReal_lt_top))] |
| simpa only [ENNReal.toReal_mul,ENNReal.toReal_ofReal (hφ0 _),smul_eq_mul] using he |
|
|
| lemma log_bounded_of_bounds {α : Type*} (f : α → ℝ) {a b : ℝ} (ha : 0 < a) |
| (hf : ∀ x,a ≤ f x ∧ f x ≤ b) : |
| ∀ x,‖Real.log (f x)‖ ≤ |Real.log a|+|Real.log b| := by |
| intro x |
| rw [Real.norm_eq_abs] |
| have hl := Real.log_le_log ha (hf x).1 |
| have hh := Real.log_le_log (ha.trans_le (hf x).1) (hf x).2 |
| apply abs_le.mpr |
| constructor |
| · have := neg_abs_le (Real.log a); have := abs_nonneg (Real.log b); linarith |
| · have := le_abs_self (Real.log b); have := abs_nonneg (Real.log a); linarith |
|
|
| lemma referenceWeight_log_bounded {d : ℕ} (hd : 16 ≤ d) (ν : Measure (Ambient d)) |
| [IsProbabilityMeasure ν] {H : ℝ} (hH : 0 < H) |
| (hν : ν ≤ ENNReal.ofReal H • sphereProbability d) (r : Ambient d → ℝ) (hr : Measurable r) |
| {ε : ℝ} (hε : 0 < ε) (hrε : ∀ s,ε ≤ r s) (hr1 : ∀ s,r s ≤ 1) : |
| ∃ M : ℝ,0 ≤ M ∧ ∀ s,‖Real.log (referenceWeight ν r s)‖ ≤ M := by |
| obtain ⟨a,b,ha,hb⟩ := referenceWeight_bounds hd ν hH hν r hr hε hrε hr1 |
| exact ⟨|Real.log a|+|Real.log b|,by positivity,log_bounded_of_bounds _ ha hb⟩ |
|
|
| end KernelPotential.Noise |
|
|
| end OAI |
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