| import Mathlib |
| import OAI.Probability.SKRatio.Gaussian.ExponentialEvents |
| import OAI.Probability.SKRatio.Gaussian.EmpiricalIntervals |
| import OAI.Probability.SKRatio.Variational.ScalarMoments |
| import OAI.Probability.SKRatio.Quantization.BinParameters |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators NNReal ENNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Real Set |
| namespace SKRatio.Bins |
| open Planted Scalar SKRatioClock.Regression |
| attribute [local instance] Classical.propDecidable |
|
|
| def intervalBin {N : ℕ} (e : Fin N → ℝ) (x : ℝ) : Fin (N+1) := |
| ⟨(Finset.univ.filter (fun i => e i<x)).card,by |
| have h := Finset.card_le_card (Finset.filter_subset (s:=Finset.univ) (p:=fun i => e i<x)) |
| simpa only [Finset.card_univ,Fintype.card_fin] using Nat.lt_succ_of_le h⟩ |
|
|
| lemma intervalBin_le_cut {N : ℕ} {e : Fin N → ℝ} (he : StrictMono e) |
| (x : ℝ) (j : Fin N) : (intervalBin e x).val≤j.val ↔ x≤e j := by |
| constructor |
| · intro hj |
| by_contra hh |
| have hx : e j<x := lt_of_not_ge hh |
| have hsub : Finset.Iic j ⊆ Finset.univ.filter (fun i => e i<x) := by |
| intro i hi |
| exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,(he.monotone (Finset.mem_Iic.mp hi)).trans_lt hx⟩ |
| have hc := Finset.card_le_card hsub |
| rw [Fin.card_Iic] at hc |
| change (Finset.univ.filter (fun i => e i<x)).card≤j.val at hj |
| omega |
| · intro hx |
| have hsub : Finset.univ.filter (fun i => e i<x) ⊆ Finset.Iio j := by |
| intro i hi |
| exact Finset.mem_Iio.mpr (he.lt_iff_lt.mp ((Finset.mem_filter.mp hi).2.trans_le hx)) |
| have hc := Finset.card_le_card hsub |
| rwa [Fin.card_Iio] at hc |
|
|
| lemma intervalBin_eq_zero {N : ℕ} (hN : 0<N) {e : Fin N → ℝ} (he : StrictMono e) (x : ℝ) : |
| intervalBin e x=0 ↔ x≤e ⟨0,hN⟩ := by |
| rw [←intervalBin_le_cut he x ⟨0,hN⟩] |
| simp only [Fin.ext_iff,Fin.val_zero,Nat.le_zero_eq] |
|
|
| lemma intervalBin_eq_last {N : ℕ} (hN : 0<N) {e : Fin N → ℝ} (he : StrictMono e) (x : ℝ) : |
| intervalBin e x=Fin.last N ↔ e ⟨N-1,by omega⟩<x := by |
| have hh := intervalBin_le_cut he x ⟨N-1,by omega⟩ |
| simp only [Fin.ext_iff,Fin.val_last] |
| have hbound := (intervalBin e x).isLt |
| constructor |
| · intro h |
| exact lt_of_not_ge (fun hx => by have := hh.mpr hx; omega) |
| · intro h |
| have hh' : ¬ (intervalBin e x).val≤N-1 := fun hu => not_le.mpr h (hh.mp hu) |
| omega |
|
|
| lemma intervalBin_eq_middle {N : ℕ} {e : Fin N → ℝ} (he : StrictMono e) |
| (x : ℝ) (j : Fin (N+1)) (hj0 : 0<j.val) (hjN : j.val<N) : |
| intervalBin e x=j ↔ e ⟨j.val-1,by omega⟩<x ∧ x≤e ⟨j.val,hjN⟩ := by |
| have hlo := intervalBin_le_cut he x ⟨j.val-1,by omega⟩ |
| have hhi := intervalBin_le_cut he x ⟨j.val,hjN⟩ |
| dsimp only [Fin.val_mk] at hlo hhi |
| rw [Fin.ext_iff] |
| constructor |
| · intro hh |
| constructor |
| · exact lt_of_not_ge (fun hx => by have := hlo.mpr hx; omega) |
| · exact hhi.mp (by omega) |
| · rintro ⟨hl,hu⟩ |
| have h1 := hhi.mpr hu |
| have h2 : ¬ (intervalBin e x).val≤j.val-1 := fun h => not_le.mpr hl (hlo.mp h) |
| omega |
|
|
| lemma measurableSet_intervalBin {N : ℕ} (hN : 0<N) {e : Fin N → ℝ} (he : StrictMono e) |
| (j : Fin (N+1)) : MeasurableSet {x | intervalBin e x=j} := by |
| by_cases hj0 : j.val=0 |
| · have hj : j=0 := Fin.ext hj0 |
| simp only [hj,intervalBin_eq_zero hN he] |
| exact measurableSet_Iic |
| · by_cases hjN : j.val=N |
| · have hj : j=Fin.last N := Fin.ext hjN |
| simp only [hj,intervalBin_eq_last hN he] |
| exact measurableSet_Ioi |
| · have h1 : 0<j.val := Nat.pos_of_ne_zero hj0 |
| have h2 : j.val<N := by have := j.isLt; omega |
| simp only [intervalBin_eq_middle he _ j h1 h2] |
| exact measurableSet_Ioc |
| |
| lemma measurable_intervalBin {N : ℕ} (hN : 0<N) {e : Fin N → ℝ} (he : StrictMono e) : |
| Measurable (intervalBin e) := by |
| apply measurable_to_countable' |
| intro j |
| exact measurableSet_intervalBin hN he j |
|
|
| lemma intervalBin_empirical {N : ℕ} (hN : 0<N) {e : Fin N → ℝ} (he : StrictMono e) |
| {Ω : ℕ → Type*} [∀ n, MeasurableSpace (Ω n)] |
| {ρ : ∀ n, Measure (Ω n)} {X : ∀ n, Ω n → Fin n → ℝ} |
| {μ : Measure ℝ} [IsProbabilityMeasure μ] [NullSingletonClass μ] |
| (hX : ExponentialEmpiricalConcentration ρ X μ) (j : Fin (N+1)) : |
| ExponentialConvergence ρ |
| (fun n ω => (count (fun i => intervalBin e (X n ω i)) j:ℝ)/(n:ℝ)) |
| (μ.real {x | intervalBin e x=j}) := by |
| have heq (n : ℕ) (ω : Ω n) : (count (fun i => intervalBin e (X n ω i)) j:ℝ) = |
| ∑ i, {x | intervalBin e x=j}.indicator (fun _ : ℝ => (1:ℝ)) (X n ω i) := by |
| simp only [count,Finset.cast_card,Set.indicator,Set.mem_ofPred_eq] |
| exact Finset.sum_filter _ _ |
| simp_rw [heq] |
| by_cases hj0 : j.val=0 |
| · have hj : j=0 := Fin.ext hj0 |
| simpa only [hj,intervalBin_eq_zero hN he,Set.Iic_def] using hX.Iic (e ⟨0,hN⟩) |
| · by_cases hjN : j.val=N |
| · have hj : j=Fin.last N := Fin.ext hjN |
| simpa only [hj,intervalBin_eq_last hN he,Set.Ioi_def] using hX.Ioi (e ⟨N-1,by omega⟩) |
| · have h1 : 0<j.val := Nat.pos_of_ne_zero hj0 |
| have h2 : j.val<N := by have := j.isLt; omega |
| simpa only [intervalBin_eq_middle he _ j h1 h2,Set.Ioc_def] using |
| hX.Ioc (he.monotone (show (⟨j.val-1,by omega⟩ : Fin N)≤⟨j.val,h2⟩ from by simp)) |
|
|
| lemma gaussianReal_real_pos_of_volume_pos (μ : ℝ) {s : ℝ≥0} (hs : s≠0) |
| {E : Set ℝ} (hE : 0<volume E) : 0<(gaussianReal μ s).real E := by |
| apply ENNReal.toReal_pos _ (measure_ne_top _ _) |
| exact fun h => hE.ne' (gaussianReal_absolutelyContinuous' μ hs h) |
|
|
| lemma intervalBin_mass_pos {N : ℕ} (hN : 0<N) {e : Fin N → ℝ} (he : StrictMono e) |
| (β : ℝ) (hβ : β≠0) (j : Fin (N+1)) : |
| 0<(fieldLaw β).real {x | intervalBin e x=j} := by |
| have hv : variance β≠0 := by |
| intro hh |
| have hh' := congrArg (fun x : ℝ≥0 => (x:ℝ)) hh |
| rw [variance_coe,NNReal.coe_zero] at hh' |
| exact hβ (sq_eq_zero_iff.mp hh') |
| apply gaussianReal_real_pos_of_volume_pos (β^2) hv |
| by_cases hj0 : j.val=0 |
| · have hj : j=0 := Fin.ext hj0 |
| simp only [hj,intervalBin_eq_zero hN he,Set.Iic_def,Real.volume_Iic] |
| exact ENNReal.zero_lt_top |
| · by_cases hjN : j.val=N |
| · have hj : j=Fin.last N := Fin.ext hjN |
| simp only [hj,intervalBin_eq_last hN he,Set.Ioi_def,Real.volume_Ioi] |
| exact ENNReal.zero_lt_top |
| · have h1 : 0<j.val := Nat.pos_of_ne_zero hj0 |
| have h2 : j.val<N := by have := j.isLt; omega |
| simp only [intervalBin_eq_middle he _ j h1 h2] |
| change 0<volume (Ioc _ _) |
| rw [Real.volume_Ioc,ENNReal.ofReal_pos] |
| exact sub_pos.mpr (he (show (⟨j.val-1,by omega⟩ : Fin N)<⟨j.val,h2⟩ from by change j.val-1<j.val; omega)) |
| |
| end SKRatio.Bins |
| |
| end |
| |
| end OAI |
| |
