| import Mathlib |
| import OAI.Probability.Perceptron.Cascade.IndexedMarkedPoisson |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace SphericalPerceptronFreeEnergy |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal BigOperators |
|
|
| lemma indexedPoissonMeasure_lintegral {A : Type*} [MeasurableSpace A] |
| (a : ℕ → ℕ × (ℕ → A)) {f : A → ℝ≥0∞} (hf : Measurable f) : |
| (∫⁻ x, f x ∂indexedPoissonMeasure a) = |
| ∑' i, ∑ j : Fin ((a i).1), f ((a i).2 j.val) := by |
| simp only [indexedPoissonMeasure,indexedFiniteCloud,finitePointMeasure, |
| lintegral_sum_measure,lintegral_finsetSum_measure,lintegral_dirac' _ hf] |
|
|
| def indexedCascadeRootMass {n : ℕ} (b : IndexedCascadeBase (n+1)) : ℝ≥0∞ := |
| markedStableTotalE (indexedPoissonMeasure b) |
|
|
| lemma indexedCascadeRootMass_measurable (n : ℕ) : |
| Measurable (indexedCascadeRootMass (n := n)) := |
| markedStableTotalE_measurable.comp indexedPoissonMeasure_measurable |
|
|
| lemma indexedCascadeRealize_rootMass {S : Type} [MeasurableSpace S] (n : ℕ) |
| (b : IndexedCascadeBase (n+1)) (m : IndexedCascadeMarks S (n+1)) : |
| markedStableTotalE (indexedCascadeRealize (n+1) (b,m)) = indexedCascadeRootMass b := by |
| unfold indexedCascadeRealize indexedCascadeRootMass markedStableTotalE |
| rw [indexedPoissonMeasure_lintegral _ (by fun_prop),indexedPoissonMeasure_lintegral _ (by fun_prop)] |
|
|
| def IndexedCascadeGood : (n : ℕ) → IndexedCascadeBase n → Prop |
| | 0,_ => True |
| | n+1,b => 0 < (indexedCascadeRootMass b).toReal ∧ |
| ∀ i j : ℕ, IndexedCascadeGood n (((b i).2 j).2) |
|
|
| lemma indexedCascadeBase_child_law (n : ℕ) (z : Fin (n+1) → ℝ) (i j : ℕ) : |
| MeasurePreserving (fun b : IndexedCascadeBase (n+1) => (((b i).2 j).2)) |
| (indexedCascadeBaseLaw (n+1) z) (indexedCascadeBaseLaw n (fun l => z l.succ)) := by |
| let B := (indexedCascadeBaseLaw n (fun l => z l.succ) : Measure (IndexedCascadeBase n)) |
| let ρ := fun i => finiteIntensityMarks (sfiniteSeq (stableLogIntensity (z 0)) i) |
| let r := fun i => ((sfiniteSeq (stableLogIntensity (z 0)) i) univ).toNNReal |
| exact (measurePreserving_snd (μ := ρ i) (ν := B)).comp |
| ((measurePreserving_eval_infinitePi (fun _ : ℕ => (ρ i).prod B) j).comp |
| ((measurePreserving_snd (μ := poissonMeasure (r i)) |
| (ν := Measure.infinitePi (fun _ : ℕ => (ρ i).prod B))).comp |
| (measurePreserving_eval_infinitePi (fun i => (poissonMeasure (r i)).prod |
| (Measure.infinitePi (fun _ : ℕ => (ρ i).prod B))) i))) |
|
|
| lemma indexedCascadeRootMass_pos (n : ℕ) (z : Fin (n+1) → ℝ) |
| (hz0 : 0 < z 0) (hz1 : z 0 < 1) : |
| ∀ᵐ b ∂(indexedCascadeBaseLaw (n+1) z : Measure (IndexedCascadeBase (n+1))), |
| 0 < (indexedCascadeRootMass b).toReal := by |
| let B := (indexedCascadeBaseLaw n (fun l => z l.succ) : Measure (IndexedCascadeBase n)) |
| let κ := stableLogIntensity (z 0) |
| have hl : (indexedCascadeBaseLaw (n+1) z : Measure (IndexedCascadeBase (n+1))).map |
| indexedPoissonMeasure = poissonRandomMeasureLaw (κ.prod B) := by |
| change (Measure.infinitePi (fun i => (poissonMeasure (((sfiniteSeq κ i) univ).toNNReal)).prod |
| (Measure.infinitePi (fun _ : ℕ => (finiteIntensityMarks (sfiniteSeq κ i)).prod B)))).map |
| indexedPoissonMeasure = _ |
| rw [indexedPoissonMeasure_law,countablePoissonLaw_product_intensity] |
| have hreg := (markedStable_regular B hz0 hz1).mono fun η hη => hη.2 |
| rw [← hl] at hreg |
| exact (ae_map_iff indexedPoissonMeasure_measurable.aemeasurable |
| (measurableSet_lt measurable_const markedStableTotal_measurable)).mp hreg |
|
|
| lemma indexedCascadeGood_ae (n : ℕ) (z : Fin n → ℝ) |
| (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1) : |
| ∀ᵐ b ∂(indexedCascadeBaseLaw n z : Measure (IndexedCascadeBase n)), IndexedCascadeGood n b := by |
| induction n with |
| | zero => exact ae_of_all _ fun _ => trivial |
| | succ n ih => |
| have ht (i j : ℕ) : ∀ᵐ b ∂(indexedCascadeBaseLaw (n+1) z : Measure (IndexedCascadeBase (n+1))), |
| IndexedCascadeGood n (((b i).2 j).2) := |
| (indexedCascadeBase_child_law n z i j).quasiMeasurePreserving.ae |
| (ih (fun l => z l.succ) (fun l => hz0 l.succ) (fun l => hz1 l.succ)) |
| filter_upwards [indexedCascadeRootMass_pos n z (hz0 0) (hz1 0), |
| ae_all_iff.mpr (fun i => ae_all_iff.mpr (ht i))] with b hb ht |
| exact ⟨hb,ht⟩ |
|
|
| lemma indexedCascadeGood_countKernel {S : Type} [MeasurableSpace S] {n : ℕ} |
| {b : IndexedCascadeBase (n+1)} (hb : IndexedCascadeGood (n+1) b) |
| (m : IndexedCascadeMarks S (n+1)) : |
| markedStableCountKernel (indexedCascadeRealize (n+1) (b,m)) = |
| indexedCascadeRealize (n+1) (b,m) := by |
| apply markedStableCountKernel_eq |
| · rw [indexedCascadeRealize_rootMass] |
| exact (ENNReal.toReal_pos_iff.mp hb.1).2.ne |
| · change 0 < (markedStableTotalE (indexedCascadeRealize (n+1) (b,m))).toReal |
| rw [indexedCascadeRealize_rootMass] |
| exact hb.1 |
|
|
| @[reducible] def IndexedLeaf : ℕ → MeasCat |
| | 0 => MeasCat.of Unit |
| | n+1 => MeasCat.of (ℕ × ℕ × IndexedLeaf n) |
|
|
| instance indexedLeaf_countable (n : ℕ) : Countable (IndexedLeaf n) := by |
| induction n with |
| | zero => dsimp [IndexedLeaf]; infer_instance |
| | succ n ih => dsimp [IndexedLeaf]; infer_instance |
|
|
| instance indexedLeaf_nonempty (n : ℕ) : Nonempty (IndexedLeaf n) := by |
| induction n with |
| | zero => exact ⟨()⟩ |
| | succ n ih => dsimp [IndexedLeaf]; infer_instance |
|
|
| instance indexedLeaf_measurableSingleton (n : ℕ) : MeasurableSingletonClass (IndexedLeaf n) := by |
| induction n with |
| | zero => dsimp [IndexedLeaf]; infer_instance |
| | succ n ih => dsimp [IndexedLeaf]; infer_instance |
|
|
| def indexedLeafWeight : (n : ℕ) → IndexedCascadeBase n → IndexedLeaf n → ℝ≥0 |
| | 0,_,_ => 1 |
| | n+1,b,l => if l.2.1 < (b l.1).1 then |
| (Real.exp (((b l.1).2 l.2.1).1)).toNNReal * |
| indexedLeafWeight n (((b l.1).2 l.2.1).2) l.2.2 else 0 |
|
|
| lemma indexedLeafWeight_measurable (n : ℕ) : |
| Measurable (Function.uncurry (indexedLeafWeight n)) := by |
| induction n with |
| | zero => exact measurable_const |
| | succ n ih => |
| apply measurable_from_prod_countable_left |
| intro l |
| change Measurable (fun b : ℕ → ℕ × (ℕ → ℝ × IndexedCascadeBase n) => |
| if l.2.1 < (b l.1).1 then (Real.exp (((b l.1).2 l.2.1).1)).toNNReal * |
| indexedLeafWeight n (((b l.1).2 l.2.1).2) l.2.2 else 0) |
| apply Measurable.ite (measurableSet_lt measurable_const (by fun_prop)) |
| · exact (by fun_prop : Measurable (fun b : ℕ → ℕ × (ℕ → ℝ × IndexedCascadeBase n) => |
| (Real.exp (((b l.1).2 l.2.1).1)).toNNReal)).mul |
| (ih.comp (show Measurable (fun b : ℕ → ℕ × (ℕ → ℝ × IndexedCascadeBase n) => |
| ((((b l.1).2 l.2.1).2),l.2.2)) from by fun_prop)) |
| · exact measurable_const |
|
|
| def indexedLeafMeasure (n : ℕ) (b : IndexedCascadeBase n) : Measure (IndexedLeaf n) := |
| Measure.sum (fun l => (indexedLeafWeight n b l : ℝ≥0∞) • Measure.dirac l) |
|
|
| instance indexedLeafMeasure_sfinite (n : ℕ) (b : IndexedCascadeBase n) : |
| SFinite (indexedLeafMeasure n b) := by |
| unfold indexedLeafMeasure |
| infer_instance |
|
|
| lemma indexedLeafMeasure_measurable (n : ℕ) : Measurable (indexedLeafMeasure n) := by |
| apply Measure.measurable_of_measurable_coe |
| intro s hs |
| simp only [indexedLeafMeasure,Measure.sum_apply _ hs,Measure.smul_apply] |
| apply Measurable.tsum |
| intro l |
| exact (((indexedLeafWeight_measurable n).comp (measurable_id.prodMk measurable_const)).coe_nnreal_ennreal).mul_const _ |
|
|
| lemma indexedLeafMeasure_lintegral (n : ℕ) (b : IndexedCascadeBase n) (f : IndexedLeaf n → ℝ≥0∞) : |
| (∫⁻ l, f l ∂indexedLeafMeasure n b) = ∑' l, (indexedLeafWeight n b l : ℝ≥0∞)*f l := by |
| simp only [indexedLeafMeasure,lintegral_sum_measure,lintegral_smul_measure,lintegral_dirac,smul_eq_mul] |
|
|
| lemma indexedLeafMeasure_lintegral_succ (n : ℕ) (b : IndexedCascadeBase (n+1)) |
| (f : IndexedLeaf (n+1) → ℝ≥0∞) : |
| (∫⁻ l, f l ∂indexedLeafMeasure (n+1) b) = |
| ∑' i, ∑ j : Fin ((b i).1), ENNReal.ofReal (Real.exp (((b i).2 j.val).1))* |
| ∫⁻ l, f (i,j.val,l) ∂indexedLeafMeasure n (((b i).2 j.val).2) := by |
| rw [indexedLeafMeasure_lintegral] |
| change (∑' l : ℕ × ℕ × IndexedLeaf n, (indexedLeafWeight (n+1) b l : ℝ≥0∞)*f l) = _ |
| rw [ENNReal.tsum_prod'] |
| apply tsum_congr |
| intro i |
| rw [ENNReal.tsum_prod'] |
| have he (j : ℕ) : (∑' l : IndexedLeaf n, |
| (indexedLeafWeight (n+1) b (i,j,l) : ℝ≥0∞)*f (i,j,l)) = |
| if j < (b i).1 then ENNReal.ofReal (Real.exp (((b i).2 j).1))* |
| (∫⁻ l, f (i,j,l) ∂indexedLeafMeasure n (((b i).2 j).2)) else 0 := by |
| by_cases hj : j < (b i).1 |
| · simp only [indexedLeafWeight,ite_eq_left hj,ENNReal.coe_mul] |
| change (∑' l : IndexedLeaf n, ENNReal.ofReal (Real.exp (((b i).2 j).1)) * |
| (indexedLeafWeight n (((b i).2 j).2) l : ℝ≥0∞)*f (i,j,l)) = _ |
| rw [indexedLeafMeasure_lintegral,← ENNReal.tsum_mul_left] |
| exact tsum_congr (fun l => mul_assoc _ _ _) |
| · simp [indexedLeafWeight,hj] |
| simp_rw [he] |
| rw [tsum_eq_sum (s := Finset.range ((b i).1)) (fun j hj => by simp [Finset.mem_range.not.mp hj])] |
| rw [Fin.sum_univ_eq_sum_range (fun j => ENNReal.ofReal (Real.exp (((b i).2 j).1)) * |
| ∫⁻ l, f (i,j,l) ∂indexedLeafMeasure n (((b i).2 j).2))] |
| apply Finset.sum_congr rfl |
| intro j hj |
| rw [ite_eq_left (Finset.mem_range.mp hj)] |
|
|
| def indexedLeafState {X S : Type} [MeasurableSpace X] [MeasurableSpace S] |
| (step : X × S → X) : (n : ℕ) → X × IndexedCascadeMarks S n → IndexedLeaf n → X |
| | 0,p,_ => p.1 |
| | n+1,p,l => indexedLeafState step n (step (p.1,(p.2 l.1 l.2.1).1), |
| (p.2 l.1 l.2.1).2) l.2.2 |
|
|
| lemma indexedLeafState_measurable {X S : Type} [MeasurableSpace X] [MeasurableSpace S] |
| {step : X × S → X} (hs : Measurable step) (n : ℕ) : |
| Measurable (Function.uncurry (indexedLeafState step n)) := by |
| induction n with |
| | zero => exact measurable_fst.fst |
| | succ n ih => |
| apply measurable_from_prod_countable_left |
| intro l |
| change Measurable (fun p : X × (ℕ → ℕ → S × IndexedCascadeMarks S n) => |
| indexedLeafState step n (step (p.1,(p.2 l.1 l.2.1).1),(p.2 l.1 l.2.1).2) l.2.2) |
| exact ih.comp (show Measurable (fun p : X × (ℕ → ℕ → S × IndexedCascadeMarks S n) => |
| ((step (p.1,(p.2 l.1 l.2.1).1),(p.2 l.1 l.2.1).2),l.2.2)) from by fun_prop) |
|
|
| end SphericalPerceptronFreeEnergy |
|
|
| end |
|
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| end OAI |
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