| import OAI.Probability.GaussianPropeller.HeatConvolution |
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| namespace OAI |
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| universe uX |
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| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| open scoped RealInnerProductSpace |
| open scoped RealInnerProductSpace |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal RealInnerProductSpace |
| open Filter |
| open scoped Topology |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Topology |
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| namespace GaussianPropeller.Quantile |
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| noncomputable def φ (x : ℝ) : ℝ := (Real.sqrt (2*Real.pi))⁻¹ * Real.exp (-x^2/2) |
| noncomputable def Φ (x : ℝ) : ℝ := ∫ y in Iic x, φ y |
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| lemma φ_pos (x : ℝ) : 0 < φ x := by unfold φ; positivity |
| lemma φ_neg (x : ℝ) : φ (-x) = φ x := by simp [φ] |
| lemma integrable_φ : Integrable φ := by |
| unfold φ |
| have h := integrable_exp_neg_mul_sq (by norm_num : (0:ℝ) < 1/2) |
| simpa only [φ, show ∀ x : ℝ, -x^2/2 = -(1/2:ℝ)*x^2 by intro x; ring] using |
| h.const_mul ((Real.sqrt (2*Real.pi))⁻¹) |
| lemma integral_φ : ∫ x, φ x = 1 := by |
| unfold φ |
| rw [integral_const_mul] |
| simp_rw [show ∀ x : ℝ, -x^2/2 = -(1/2:ℝ)*x^2 by intro x; ring] |
| rw [integral_gaussian] |
| have hp : Real.pi/(1/2:ℝ) = 2*Real.pi := by ring |
| rw [hp, inv_mul_cancel₀ (by positivity)] |
| lemma hasDerivAt_φ (x : ℝ) : HasDerivAt φ (-x*φ x) x := by |
| unfold φ |
| convert ((((hasDerivAt_pow 2 x).neg.div_const 2).exp).const_mul |
| ((Real.sqrt (2*Real.pi))⁻¹)) using 1 |
| first | rfl | (dsimp; ring) |
| lemma continuous_φ : Continuous φ := by exact continuous_iff_continuousAt.mpr fun x => |
| (hasDerivAt_φ x).continuousAt |
| lemma hasDerivAt_Φ (x : ℝ) : HasDerivAt Φ (φ x) x := by |
| have heq : Φ = fun a => Φ 0 + ∫ t in (0:ℝ)..a, φ t := by |
| funext a |
| have hi := intervalIntegral.integral_Iic_sub_Iic |
| (a := 0) (b := a) integrable_φ.integrableOn integrable_φ.integrableOn |
| change Φ a-Φ 0 = _ at hi |
| linarith only [hi] |
| rw [heq] |
| exact (intervalIntegral.integral_hasDerivAt_right integrable_φ.intervalIntegrable |
| continuous_φ.stronglyMeasurable.stronglyMeasurableAtFilter continuous_φ.continuousAt).const_add _ |
| lemma continuous_Φ : Continuous Φ := continuous_iff_continuousAt.mpr fun x => |
| (hasDerivAt_Φ x).continuousAt |
| lemma strictMono_Φ : StrictMono Φ := strictMono_of_deriv_pos (fun x => by |
| rw [(hasDerivAt_Φ x).deriv]; exact φ_pos x) |
| lemma Φ_pos (x : ℝ) : 0 < Φ x := by |
| apply (setIntegral_pos_iff_support_of_nonneg_ae (ae_of_all _ (fun x => (φ_pos x).le)) |
| integrable_φ.integrableOn).mpr |
| simp [Function.support, ne_of_gt (φ_pos _)] |
| lemma Φ_eq_one_sub_tail (x : ℝ) : Φ x = 1 - ∫ y in Ioi x, φ y := by |
| have h := integral_add_compl measurableSet_Iic integrable_φ (s := Iic x) |
| rw [compl_Iic, integral_φ] at h |
| unfold Φ |
| linarith only [h] |
| lemma Φ_neg (x : ℝ) : Φ (-x) = 1-Φ x := by |
| have h := integral_comp_neg_Iic (-x) φ |
| simp only [φ_neg, neg_neg] at h |
| change Φ (-x) = (∫ y in Ioi x, φ y) at h |
| rw [h, Φ_eq_one_sub_tail] |
| ring |
| lemma Φ_lt_one (x : ℝ) : Φ x < 1 := by |
| have h := Φ_pos (-x) |
| rw [Φ_neg] at h |
| linarith only [h] |
| lemma Φ_mem (x : ℝ) : Φ x ∈ Ioo (0:ℝ) 1 := ⟨Φ_pos x, Φ_lt_one x⟩ |
| lemma Φ_zero : Φ 0 = 1/2 := by |
| have h := Φ_neg 0 |
| simp only [neg_zero] at h |
| linarith only [h] |
| lemma tendsto_Φ_atTop : Tendsto Φ atTop (𝓝 1) := by |
| have ht := tendsto_integral_Ioi_zero (f := φ) (μ := volume) tendsto_id |
| convert ht.const_sub 1 using 1 |
| · exact funext Φ_eq_one_sub_tail |
| · norm_num |
| lemma tendsto_Φ_atBot : Tendsto Φ atBot (𝓝 0) := by |
| have ht := tendsto_Φ_atTop.comp tendsto_neg_atBot_atTop |
| have hh := ht.const_sub 1 |
| simpa only [Function.comp_def, Φ_neg, sub_sub_cancel, sub_self] using hh |
| lemma range_Φ : range Φ = Ioo (0:ℝ) 1 := by |
| apply Subset.antisymm |
| · rintro _ ⟨x,rfl⟩; exact Φ_mem x |
| · have hh := isPreconnected_univ.intermediate_value_Ioo |
| (l₁ := atBot) (l₂ := atTop) (by simp) (by simp) continuous_Φ.continuousOn |
| tendsto_Φ_atBot tendsto_Φ_atTop |
| simpa only [image_univ] using hh |
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| noncomputable def cdfIso : ℝ ≃o Ioo (0:ℝ) 1 := |
| (strictMono_Φ.orderIso Φ).trans (Set.orderIsoOfEq _ _ range_Φ) |
| lemma cdfIso_coe (x : ℝ) : (cdfIso x : ℝ) = Φ x := rfl |
| noncomputable def q (p : ℝ) : ℝ := |
| if h : p ∈ Ioo (0:ℝ) 1 then cdfIso.symm ⟨p,h⟩ else 0 |
| lemma q_of_mem {p : ℝ} (hp : p ∈ Ioo (0:ℝ) 1) : q p = cdfIso.symm ⟨p,hp⟩ := by |
| simp [q, hp] |
| lemma Φ_q {p : ℝ} (hp : p ∈ Ioo (0:ℝ) 1) : Φ (q p) = p := by |
| rw [q_of_mem hp, ← cdfIso_coe] |
| exact congrArg Subtype.val (cdfIso.apply_symm_apply ⟨p,hp⟩) |
| lemma q_Φ (x : ℝ) : q (Φ x) = x := by |
| rw [q_of_mem (Φ_mem x)] |
| exact cdfIso.symm_apply_apply x |
| lemma continuousOn_q : ContinuousOn q (Ioo (0:ℝ) 1) := by |
| rw [continuousOn_iff_continuous_domRestrict] |
| change Continuous (fun p : Ioo (0:ℝ) 1 => q p) |
| have heq : (fun p : Ioo (0:ℝ) 1 => q p) = cdfIso.symm := by |
| funext p |
| exact q_of_mem p.2 |
| rw [heq] |
| exact cdfIso.symm.continuous |
| lemma continuousAt_q {p : ℝ} (hp : p ∈ Ioo (0:ℝ) 1) : ContinuousAt q p := |
| continuousOn_q.continuousAt (isOpen_Ioo.mem_nhds hp) |
| lemma hasDerivAt_q {p : ℝ} (hp : p ∈ Ioo (0:ℝ) 1) : |
| HasDerivAt q (φ (q p))⁻¹ p := by |
| apply HasDerivAt.of_local_left_inverse (continuousAt_q hp) (hasDerivAt_Φ (q p)) |
| (ne_of_gt (φ_pos _)) |
| filter_upwards [isOpen_Ioo.mem_nhds hp] with y hy |
| exact Φ_q hy |
| lemma strictMonoOn_q : StrictMonoOn q (Ioo (0:ℝ) 1) := by |
| intro a ha b hb hab |
| rw [q_of_mem ha, q_of_mem hb] |
| exact cdfIso.symm.strictMono hab |
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| noncomputable def I (p : ℝ) : ℝ := φ (q p) |
| lemma I_pos (p : ℝ) : 0 < I p := φ_pos _ |
| lemma I_Φ (x : ℝ) : I (Φ x) = φ x := by simp [I, q_Φ] |
| lemma hasDerivAt_I {p : ℝ} (hp : p ∈ Ioo (0:ℝ) 1) : HasDerivAt I (-q p) p := by |
| convert (hasDerivAt_φ (q p)).comp p (hasDerivAt_q hp) using 1 <;> |
| first | rfl | field_simp [ne_of_gt (φ_pos (q p))] |
| lemma antitoneOn_I_deriv : AntitoneOn (fun p => -q p) (Ioo (0:ℝ) 1) := by |
| intro a ha b hb hab |
| exact neg_le_neg (strictMonoOn_q.monotoneOn ha hb hab) |
| lemma concaveOn_I : ConcaveOn ℝ (Ioo (0:ℝ) 1) I := by |
| have hanti : AntitoneOn (deriv I) (Ioo (0:ℝ) 1) := by |
| intro a ha b hb hab |
| rw [(hasDerivAt_I ha).deriv, (hasDerivAt_I hb).deriv] |
| exact antitoneOn_I_deriv ha hb hab |
| apply (hanti.mono interior_subset).concaveOn_of_deriv (convex_Ioo 0 1) |
| · exact fun p hp => (hasDerivAt_I hp).continuousAt.continuousWithinAt |
| · exact fun p hp => (hasDerivAt_I (interior_subset hp)).differentiableAt.differentiableWithinAt |
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| lemma concave_le_tangent {S : Set ℝ} {f : ℝ → ℝ} (hf : ConcaveOn ℝ S f) |
| {x y d : ℝ} (hx : x ∈ S) (hy : y ∈ S) (hd : HasDerivAt f d x) : |
| f y ≤ f x+d*(y-x) := by |
| rcases lt_trichotomy x y with h | rfl | h |
| · have hh := hf.slope_le_of_hasDerivAt hx hy h hd |
| rw [slope_def_field] at hh |
| have hh' := (div_le_iff₀ (sub_pos.mpr h)).mp hh |
| linarith |
| · simp |
| · have hh := hf.le_slope_of_hasDerivAt hy hx h hd |
| rw [slope_def_field] at hh |
| have hh' := (le_div_iff₀ (sub_pos.mpr h)).mp hh |
| nlinarith only [hh'] |
| |
| lemma I_le_tangent {p r : ℝ} (hp : p ∈ Ioo (0:ℝ) 1) (hr : r ∈ Ioo (0:ℝ) 1) : |
| I r ≤ I p-q p*(r-p) := by |
| simpa only [neg_mul, sub_eq_add_neg] using concave_le_tangent concaveOn_I hp hr (hasDerivAt_I hp) |
| |
| lemma φ_le_zero (x : ℝ) : φ x ≤ φ 0 := by |
| unfold φ |
| simp only [zero_pow (by norm_num : 2 ≠ 0), neg_zero, zero_div, Real.exp_zero, mul_one] |
| exact mul_le_of_le_one_right (by positivity) (Real.exp_le_one_iff.mpr (by nlinarith [sq_nonneg x])) |
| |
| section Integration |
| variable {X : Type uX} [MeasurableSpace X] {μ : Measure X} [IsProbabilityMeasure μ] |
| |
| lemma integrable_Φ_comp {f : X → ℝ} (hf : AEStronglyMeasurable f μ) : |
| Integrable (fun x => Φ (f x)) μ := by |
| apply Integrable.mono' (integrable_const (1:ℝ)) (continuous_Φ.comp_aestronglyMeasurable hf) |
| filter_upwards [] with x |
| rw [Real.norm_eq_abs, abs_of_pos (Φ_pos _)] |
| exact (Φ_lt_one _).le |
| lemma integrable_φ_comp {f : X → ℝ} (hf : AEStronglyMeasurable f μ) : |
| Integrable (fun x => φ (f x)) μ := by |
| apply Integrable.mono' (integrable_const (φ 0)) (continuous_φ.comp_aestronglyMeasurable hf) |
| filter_upwards [] with x |
| rw [Real.norm_eq_abs, abs_of_pos (φ_pos _)] |
| exact φ_le_zero _ |
| lemma integral_Φ_mem {f : X → ℝ} (hf : AEStronglyMeasurable f μ) : |
| (∫ x, Φ (f x) ∂μ) ∈ Ioo (0:ℝ) 1 := by |
| have hpos {g : X → ℝ} (hg : Integrable g μ) (hh : ∀ x, 0 < g x) : 0 < ∫ x, g x ∂μ := by |
| apply (integral_pos_iff_support_of_nonneg (fun x => (hh x).le) hg).mpr |
| have heq : Function.support g = univ := by ext x; simp [Function.mem_support, ne_of_gt (hh x)] |
| simp [heq] |
| refine ⟨hpos (integrable_Φ_comp hf) (fun x => Φ_pos _), ?_⟩ |
| have hh := hpos ((integrable_const (1:ℝ)).sub (integrable_Φ_comp hf)) |
| (fun x => sub_pos.mpr (Φ_lt_one (f x))) |
| simp only [Pi.sub_apply] at hh |
| rw [integral_sub (integrable_const (1:ℝ)) (integrable_Φ_comp hf), integral_const] at hh |
| simp only [probReal_univ, smul_eq_mul, one_mul] at hh |
| linarith only [hh] |
| |
| lemma integral_φ_le_I {f : X → ℝ} (hf : AEStronglyMeasurable f μ) : |
| (∫ x, φ (f x) ∂μ) ≤ I (∫ x, Φ (f x) ∂μ) := by |
| let p := ∫ x, Φ (f x) ∂μ |
| have hp : p ∈ Ioo (0:ℝ) 1 := integral_Φ_mem hf |
| have hi := integral_mono (integrable_φ_comp hf) |
| ((integrable_const (I p)).sub (((integrable_Φ_comp hf).sub (integrable_const p)).const_mul (q p))) |
| (fun x => by simpa only [I_Φ, Pi.sub_apply] using I_le_tangent hp (Φ_mem (f x))) |
| simp only [Pi.sub_apply] at hi |
| rw [integral_sub (integrable_const (I p)), integral_const, integral_const_mul, |
| integral_sub (integrable_Φ_comp hf) (integrable_const p), integral_const] at hi |
| · simpa only [probReal_univ, smul_eq_mul, one_mul, show (∫ x, Φ (f x) ∂μ) = p from rfl, |
| sub_self, mul_zero, sub_zero] using hi |
| · exact ((integrable_Φ_comp hf).sub (integrable_const p)).const_mul (q p) |
| |
| lemma hasDerivAt_integral_shift {f : X → ℝ} (hf : AEStronglyMeasurable f μ) (a : ℝ) : |
| HasDerivAt (fun s => ∫ x, Φ (f x+s) ∂μ) (∫ x, φ (f x+a) ∂μ) a := by |
| apply (hasDerivAt_integral_of_dominated_loc_of_deriv_le (s := univ) (bound := fun _ => φ 0) |
| (F' := fun s x => φ (f x+s)) (by simp) ?_ (integrable_Φ_comp (hf.add_const a)) |
| (continuous_φ.comp_aestronglyMeasurable (hf.add_const a)) ?_ (integrable_const (φ 0)) ?_).2 |
| · exact Filter.Eventually.of_forall fun s => continuous_Φ.comp_aestronglyMeasurable (hf.add_const s) |
| · filter_upwards [] with x |
| intro s _ |
| rw [Real.norm_eq_abs, abs_of_pos (φ_pos _)] |
| exact φ_le_zero _ |
| · filter_upwards [] with x |
| intro s _ |
| simpa only [mul_one, Function.comp_def, id_eq] using (hasDerivAt_Φ (f x+s)).comp s ((hasDerivAt_id s).const_add (f x)) |
|
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| lemma hasDerivAt_quantile_shift {f : X → ℝ} (hf : AEStronglyMeasurable f μ) (a : ℝ) : |
| HasDerivAt (fun s => q (∫ x, Φ (f x+s) ∂μ)) |
| ((∫ x, φ (f x+a) ∂μ)/I (∫ x, Φ (f x+a) ∂μ)) a := by |
| convert (hasDerivAt_q (integral_Φ_mem (hf.add_const a))).comp a |
| (hasDerivAt_integral_shift hf a) using 1 <;> first | rfl | (simp only [I]; ring) |
|
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| lemma quantile_shift_le {f : X → ℝ} (hf : AEStronglyMeasurable f μ) {a : ℝ} (ha : 0 ≤ a) : |
| q (∫ x, Φ (f x+a) ∂μ) ≤ q (∫ x, Φ (f x) ∂μ)+a := by |
| have hd (s : ℝ) := (hasDerivAt_quantile_shift hf s).sub (hasDerivAt_id s) |
| have hm : Antitone (fun s => q (∫ x, Φ (f x+s) ∂μ)-s) := antitone_of_hasDerivAt_nonpos hd |
| (fun s => by |
| have hh := (div_le_one (I_pos _)).mpr (integral_φ_le_I (hf.add_const s)) |
| exact sub_nonpos.mpr hh) |
| have hh := hm ha |
| simp only [add_zero, sub_zero] at hh |
| linarith only [hh] |
|
|
| end Integration |
| end GaussianPropeller.Quantile |
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| end OAI |
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