| import Mathlib |
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| namespace OAI |
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| section |
| noncomputable section |
| open scoped BigOperators NNReal ENNReal Topology |
| open MeasureTheory ProbabilityTheory Filter |
| namespace SKRatioClock.Regression |
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| def ExponentialEmpiricalConcentration {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| {E : Type*} [PseudoMetricSpace E] [MeasurableSpace E] |
| (ρ : ∀ n, Measure (H n)) (X : ∀ n, H n → Fin n → E) (ν : Measure E) : Prop := |
| ∀ (f : E → ℝ) (K : ℝ≥0), LipschitzWith K f → ∀ B : ℝ, |
| (∀ x, |f x| ≤ B) → ∀ ε : ℝ, 0 < ε → |
| ∃ C c : ℝ, 0 < C ∧ 0 < c ∧ ∀ᶠ n in atTop, |
| ρ n {h | ε ≤ |(∑ i, f (X n h i))/(n:ℝ) - ∫ x, f x ∂ν|} ≤ |
| ENNReal.ofReal (C * Real.exp (-c*(n:ℝ))) |
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| def ExponentiallyRare {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| (ρ : ∀ n, Measure (H n)) (A : ∀ n, Set (H n)) : Prop := |
| ∃ C c : ℝ, 0 < C ∧ 0 < c ∧ ∀ᶠ n in atTop, |
| ρ n (A n) ≤ ENNReal.ofReal (C * Real.exp (-c*(n:ℝ))) |
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| lemma ExponentiallyRare.mono {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| {ρ : ∀ n, Measure (H n)} {A B : ∀ n, Set (H n)} |
| (hB : ExponentiallyRare ρ B) (hAB : ∀ᶠ n in atTop, A n ⊆ B n) : |
| ExponentiallyRare ρ A := by |
| obtain ⟨C,c,hC,hc,he⟩ := hB |
| refine ⟨C,c,hC,hc,?_⟩ |
| filter_upwards [he,hAB] with n hn hABn |
| exact (measure_mono hABn).trans hn |
|
|
| lemma ExponentiallyRare.union {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| {ρ : ∀ n, Measure (H n)} {A B : ∀ n, Set (H n)} |
| (hA : ExponentiallyRare ρ A) (hB : ExponentiallyRare ρ B) : |
| ExponentiallyRare ρ (fun n => A n ∪ B n) := by |
| obtain ⟨C,c,hC,hc,he⟩ := hA |
| obtain ⟨D,d,hD,hd,hf⟩ := hB |
| refine ⟨C+D,min c d,add_pos hC hD,lt_min hc hd,?_⟩ |
| filter_upwards [he,hf] with n hn hm |
| refine (measure_union_le _ _).trans ((add_le_add hn hm).trans ?_) |
| have hc' : Real.exp (-c*(n:ℝ)) ≤ Real.exp (-min c d*(n:ℝ)) := |
| Real.exp_le_exp.mpr (by |
| have := min_le_left c d |
| nlinarith [Nat.cast_nonneg (α := ℝ) n]) |
| have hd' : Real.exp (-d*(n:ℝ)) ≤ Real.exp (-min c d*(n:ℝ)) := |
| Real.exp_le_exp.mpr (by |
| have := min_le_right c d |
| nlinarith [Nat.cast_nonneg (α := ℝ) n]) |
| rw [add_mul, ENNReal.ofReal_add (by positivity) (by positivity)] |
| exact add_le_add (ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_left hc' hC.le)) |
| (ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_left hd' hD.le)) |
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| lemma exponentiallyRare_empty {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| (ρ : ∀ n, Measure (H n)) : ExponentiallyRare ρ (fun _ => ∅) := by |
| refine ⟨1,1,by norm_num,by norm_num,Filter.Eventually.of_forall ?_⟩ |
| intro n |
| simp only [measure_empty, zero_le] |
|
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| lemma ExponentiallyRare.iUnion_finite {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| {ρ : ∀ n, Measure (H n)} {ι : Type*} [Finite ι] |
| {A : ι → ∀ n, Set (H n)} (hA : ∀ i, ExponentiallyRare ρ (A i)) : |
| ExponentiallyRare ρ (fun n => ⋃ i, A i n) := by |
| classical |
| cases nonempty_fintype ι |
| have hs (s : Finset ι) : ExponentiallyRare ρ (fun n => ⋃ i ∈ s, A i n) := by |
| induction s using Finset.induction_on with |
| | empty => simpa using exponentiallyRare_empty ρ |
| | @insert i s hi ih => |
| have heq : (fun n => ⋃ j ∈ insert i s, A j n) = |
| (fun n => A i n ∪ ⋃ j ∈ s, A j n) := by |
| funext n |
| ext h |
| simp only [Set.mem_iUnion, Finset.mem_insert, Set.mem_union] |
| aesop |
| rw [heq] |
| exact (hA i).union ih |
| simpa using hs Finset.univ |
|
|
| def ExponentialConvergence {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| (ρ : ∀ n, Measure (H n)) {E : Type*} [PseudoMetricSpace E] |
| (X : ∀ n, H n → E) (x : E) : Prop := |
| ∀ ε : ℝ, 0 < ε → ExponentiallyRare ρ (fun n => {h | ε ≤ dist (X n h) x}) |
|
|
| lemma ExponentialConvergence.continuous_map |
| {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] {ρ : ∀ n, Measure (H n)} |
| {E F : Type*} [PseudoMetricSpace E] [PseudoMetricSpace F] |
| {X : ∀ n, H n → E} {x : E} (hX : ExponentialConvergence ρ X x) |
| {f : E → F} (hf : ContinuousAt f x) : |
| ExponentialConvergence ρ (fun n h => f (X n h)) (f x) := by |
| intro ε hε |
| obtain ⟨δ,hδ,hδe⟩ := Metric.continuousAt_iff.mp hf ε hε |
| apply (hX δ hδ).mono |
| refine Filter.Eventually.of_forall ?_ |
| intro n h hh |
| by_contra hnot |
| have hnear : dist (X n h) x < δ := lt_of_not_ge hnot |
| exact (not_lt_of_ge hh) (hδe hnear) |
|
|
| lemma ExponentialConvergence.prod |
| {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] {ρ : ∀ n, Measure (H n)} |
| {E F : Type*} [PseudoMetricSpace E] [PseudoMetricSpace F] |
| {X : ∀ n, H n → E} {Y : ∀ n, H n → F} {x : E} {y : F} |
| (hX : ExponentialConvergence ρ X x) (hY : ExponentialConvergence ρ Y y) : |
| ExponentialConvergence ρ (fun n h => (X n h,Y n h)) (x,y) := by |
| intro ε hε |
| apply ((hX ε hε).union (hY ε hε)).mono |
| exact Filter.Eventually.of_forall (fun n h hh => by |
| simpa only [Prod.dist_eq, le_max_iff, Set.mem_union, Set.mem_ofPred_eq] using hh) |
|
|
| lemma ExponentialConvergence.pi_finite |
| {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] {ρ : ∀ n, Measure (H n)} |
| {ι : Type*} [Fintype ι] {E : ι → Type*} [∀ i, PseudoMetricSpace (E i)] |
| {X : ∀ n, H n → ∀ i, E i} {x : ∀ i, E i} |
| (hX : ∀ i, ExponentialConvergence ρ (fun n h => X n h i) (x i)) : |
| ExponentialConvergence ρ X x := by |
| intro ε hε |
| apply (ExponentiallyRare.iUnion_finite (fun i => hX i ε hε)).mono |
| refine Filter.Eventually.of_forall ?_ |
| intro n h hh |
| by_contra hn |
| have hall : ∀ i, dist (X n h i) (x i) < ε := by |
| intro i |
| apply lt_of_not_ge |
| intro hi |
| exact hn (Set.mem_iUnion.mpr ⟨i,hi⟩) |
| exact (not_lt_of_ge hh) ((dist_pi_lt_iff hε).mpr hall) |
|
|
| lemma ExponentialEmpiricalConcentration.average |
| {E : Type*} [PseudoMetricSpace E] [MeasurableSpace E] |
| {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| {ρ : ∀ n, Measure (H n)} {X : ∀ n, H n → Fin n → E} {ν : Measure E} |
| (hX : ExponentialEmpiricalConcentration ρ X ν) |
| (f : E → ℝ) {K : ℝ≥0} (hf : LipschitzWith K f) |
| (B : ℝ) (hB : ∀ x, |f x| ≤ B) : |
| ExponentialConvergence ρ (fun n h => (∑ i, f (X n h i))/(n:ℝ)) (∫ x, f x ∂ν) := by |
| intro ε hε |
| simpa only [ExponentiallyRare, Real.dist_eq] using hX f K hf B hB ε hε |
|
|
| lemma ExponentialEmpiricalConcentration.map |
| {E F : Type*} [PseudoMetricSpace E] [MeasurableSpace E] [BorelSpace E] |
| [PseudoMetricSpace F] [MeasurableSpace F] [BorelSpace F] |
| {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| {ρ : ∀ n, Measure (H n)} {X : ∀ n, H n → Fin n → E} {ν : Measure E} |
| (hX : ExponentialEmpiricalConcentration ρ X ν) |
| (g : E → F) {L : ℝ≥0} (hg : LipschitzWith L g) : |
| ExponentialEmpiricalConcentration ρ (fun n h i => g (X n h i)) (ν.map g) := by |
| intro f K hf B hB ε hε |
| have hh := hX (f ∘ g) (K*L) (hf.comp hg) B (fun x => hB (g x)) ε hε |
| have hm : ∫ y, f y ∂ν.map g = ∫ x, f (g x) ∂ν := |
| integral_map hg.continuous.measurable.aemeasurable hf.continuous.measurable.aestronglyMeasurable |
| simpa only [Function.comp_apply, hm] using hh |
|
|
| lemma ExponentiallyRare.without_prefactor {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| {ρ : ∀ n, Measure (H n)} {A : ∀ n, Set (H n)} (hA : ExponentiallyRare ρ A) : |
| ∃ c : ℝ, 0<c ∧ ∀ᶠ n in atTop, ρ n (A n) ≤ ENNReal.ofReal (Real.exp (-c*n)) := by |
| obtain ⟨C,c,hC,hc,he⟩ := hA |
| refine ⟨c/2,by positivity,?_⟩ |
| have hnc := (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n:ℝ)) atTop atTop) |
| filter_upwards [he,hnc.eventually (eventually_ge_atTop (2*Real.log C/c))] with n hn hsize |
| apply hn.trans |
| apply ENNReal.ofReal_le_ofReal |
| rw [show C*Real.exp (-c*n)=Real.exp (Real.log C-c*n) by |
| rw [Real.exp_sub,Real.exp_log hC,neg_mul,Real.exp_neg,div_eq_mul_inv] ] |
| apply Real.exp_le_exp.mpr |
| have hh := (div_le_iff₀ hc).mp hsize |
| nlinarith |
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| end SKRatioClock.Regression |
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| end |
| end |
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| end OAI |
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