| import OAI.Probability.GaussianReplacement.Mixture |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| noncomputable section |
| namespace CurrentProjection |
| open InnerProductSpace |
| open InformationTheory |
|
|
|
|
| def cauchyObservation {d : ℕ} {C : Type*} (z : (C×Sphere d)×(Vector d×Vector d)) : |
| ((C×(Vector d×Vector d))×ℝ)×ℝ := |
| (((z.1.1,z.2),⟪(z.1.2:Vector d),z.2.1⟫),⟪(z.1.2:Vector d),z.2.2⟫) |
|
|
| lemma measurable_cauchyObservation {d : ℕ} {C : Type*} [MeasurableSpace C] : |
| Measurable (cauchyObservation (d:=d) (C:=C)) := by |
| unfold cauchyObservation |
| fun_prop |
|
|
| def cauchyObservationLaw {d : ℕ} {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) : Measure (((C×(Vector d×Vector d))×ℝ)×ℝ) := |
| (Γ.prod (gaussianVectorPair d)).map cauchyObservation |
| |
| instance cauchyObservationLaw_probability {d : ℕ} {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] : IsProbabilityMeasure (cauchyObservationLaw Γ) := by |
| exact (Measure.isProbabilityMeasure_map_iff |
| measurable_cauchyObservation.aemeasurable).mpr inferInstance |
| |
| lemma gaussian_inner_map_sphere {d : ℕ} (s : Sphere d) : |
| (stdGaussian (Vector d)).map (fun x => ⟪(s:Vector d),x⟫) = gaussianReal 0 1 := by |
| rw [gaussian_inner_map_scaled,show ‖(s:Vector d)‖ = 1 from mem_sphere_zero_iff_norm.mp s.property] |
| simp only [one_mul,Measure.map_id'] |
| |
| lemma cauchyObservationLaw_snd {d : ℕ} {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] : |
| (cauchyObservationLaw Γ).snd = gaussianReal 0 1 := by |
| rw [Measure.snd,cauchyObservationLaw,Measure.map_map measurable_snd measurable_cauchyObservation] |
| ext t ht |
| rw [Measure.map_apply (by unfold cauchyObservation; fun_prop) ht, |
| Measure.prod_apply (ht.preimage (by unfold cauchyObservation; fun_prop))] |
| have ho (z : C×Sphere d) : gaussianVectorPair d {x | ⟪(z.2:Vector d),x.2⟫ ∈ t} = gaussianReal 0 1 t := by |
| have hm : Measurable (fun x : Vector d => ⟪(z.2:Vector d),x⟫) := by fun_prop |
| have he := congrArg (fun μ : Measure ℝ => μ t) (gaussian_inner_map_sphere z.2) |
| rw [Measure.map_apply hm ht] at he |
| change gaussianVectorPair d (Prod.snd ⁻¹' ((fun x : Vector d => ⟪(z.2:Vector d),x⟫) ⁻¹' t)) = _ |
| rw [←Measure.snd_apply (ht.preimage hm),Measure.snd_prod] |
| exact he |
| change (∫⁻ z : C×Sphere d, gaussianVectorPair d {x | ⟪(z.2:Vector d),x.2⟫ ∈ t} ∂Γ) = _ |
| simp_rw [ho] |
| simp |
| |
| lemma log_standardGaussianPDF (x : ℝ) : |
| Real.log (gaussianPDFReal 0 1 x) = -Real.log (Real.sqrt (2*Real.pi))-x^2/2 := by |
| change Real.log ((Real.sqrt (2*Real.pi*1))⁻¹ * Real.exp (-(x-0)^2/(2*1))) = _ |
| simp only [mul_one,sub_zero] |
| rw [Real.log_mul (inv_ne_zero (Real.sqrt_pos.mpr (by positivity)).ne') (Real.exp_pos _).ne', |
| Real.log_inv,Real.log_exp] |
| ring |
| |
| def standardGaussianEntropy : ℝ := Real.log (Real.sqrt (2*Real.pi))+1/2 |
| |
| lemma standardGaussian_cross_entropy {T : Type*} [MeasurableSpace T] |
| (P : Measure (T×ℝ)) [IsProbabilityMeasure P] (hm : P.snd = gaussianReal 0 1) : |
| Integrable (fun z : T×ℝ => Real.log (gaussianPDFReal 0 1 z.2)) P ∧ |
| (∫ z : T×ℝ, -Real.log (gaussianPDFReal 0 1 z.2) ∂P) = standardGaussianEntropy := by |
| have hf : Measurable (fun x : ℝ => Real.log (gaussianPDFReal 0 1 x)) := by fun_prop |
| have hsq := integrable_standardNormal_square |
| have hi : Integrable (fun x : ℝ => Real.log (gaussianPDFReal 0 1 x)) (gaussianReal 0 1) := by |
| simp_rw [log_standardGaussianPDF] |
| exact (integrable_const _).sub (hsq.div_const 2) |
| refine ⟨(integrable_map_measure hf.aestronglyMeasurable measurable_snd.aemeasurable).mp (by change Integrable _ P.snd; rw [hm]; exact hi),?_⟩ |
| have he := integral_map (μ:=P) (φ:=Prod.snd) measurable_snd.aemeasurable hf.neg.aestronglyMeasurable |
| change (∫ z, (-fun x => Real.log (gaussianPDFReal 0 1 x)) z.2 ∂P) = _ |
| rw [←he] |
| change (∫ x, -Real.log (gaussianPDFReal 0 1 x) ∂P.snd) = _ |
| rw [hm] |
| simp_rw [log_standardGaussianPDF,neg_sub] |
| rw [integral_sub (hsq.div_const 2) (integrable_const _),integral_div,integral_standardNormal_square] |
| simp [standardGaussianEntropy,add_comm] |
| |
| lemma cauchyMixture_experiment_bound {d : ℕ} {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] |
| (κ : Kernel C (Sphere d)) [IsMarkovKernel κ] |
| (hfin : conditionalReferenceKL (cauchyObservationLaw Γ) (gaussianReal 0 1) ≠ ∞) |
| {e : ℝ} (he : 0 < e) (he1 : e ≤ 1) : |
| standardGaussianEntropy - (conditionalReferenceKL (cauchyObservationLaw Γ) (gaussianReal 0 1)).toReal ≤ |
| cauchyTiltConstant - ∫ z : C×Sphere d, Real.log (regularizedScale (κ z.1) e z.2) ∂Γ := by |
| let P := cauchyObservationLaw Γ |
| let Λ := P.fst.prod (volume : Measure ℝ) |
| let r : (((C×(Vector d×Vector d))×ℝ)×ℝ) → ℝ := fun z => gaussianPDFReal 0 1 z.2 |
| let q : (((C×(Vector d×Vector d))×ℝ)×ℝ) → ℝ := |
| fun z => cauchyMixture (κ z.1.1.1) e z.1.1.2 (z.1.2,z.2) |
| have hr : Measurable r := by dsimp [r]; fun_prop |
| have hq : Measurable q := (cauchyMixture_kernel_measurable κ e).comp (show Measurable (fun z : (((C×(Vector d×Vector d))×ℝ)×ℝ) => ((z.1.1.1,z.1.1.2),(z.1.2,z.2))) by fun_prop) |
| have hrpos : ∀ z, 0 < r z := fun z => gaussianPDFReal_pos _ _ _ (by norm_num) |
| have hqpos : ∀ z, 0 < q z := fun z => cauchyMixture_pos _ he _ _ |
| have hR : Λ.withDensity (fun z => ENNReal.ofReal (r z)) = P.fst.prod (gaussianReal 0 1) := by |
| rw [show (fun z => ENNReal.ofReal (r z)) = (fun z => gaussianPDF 0 1 z.2) from rfl, |
| ←prod_withDensity_right (measurable_gaussianPDF 0 1),gaussianReal_of_var_ne_zero 0 (by norm_num)] |
| have hnorm : (∫⁻ z, ENNReal.ofReal (q z) ∂Λ) = 1 := by |
| rw [lintegral_prod _ hq.ennreal_ofReal.aemeasurable] |
| simp_rw [show ∀ z : (C×(Vector d×Vector d))×ℝ, |
| (∫⁻ y, ENNReal.ofReal (q (z,y))) = 1 from fun z => cauchyMixture_normalization (κ z.1.1) he z.1.2 z.2] |
| simp |
| have hc := standardGaussian_cross_entropy P (cauchyObservationLaw_snd Γ) |
| have hqi : Integrable (fun z => Real.log (q z)) P := by |
| apply (integrable_map_measure hq.log.aestronglyMeasurable measurable_cauchyObservation.aemeasurable).mpr |
| have hi := (ownCauchyMixture_context_integrable Γ κ he he1).neg |
| convert hi using 1 |
| ext z |
| simp [q,cauchyObservation,cauchyMixture_at_own] |
| have hg := reference_gibbs P (P.fst.prod (gaussianReal 0 1)) Λ r q hr hq hrpos hqpos hR hnorm hfin hc.1 hqi |
| rw [hc.2] at hg |
| have heq : (∫ z, -Real.log (q z) ∂P) = ∫ z, -Real.log (ownCauchyMixture (κ z.1.1) e z.1.2 z.2) |
| ∂Γ.prod (gaussianVectorPair d) := by |
| rw [show P = (Γ.prod (gaussianVectorPair d)).map cauchyObservation from rfl, |
| integral_map (f:=fun z => -Real.log (q z)) measurable_cauchyObservation.aemeasurable hq.log.neg.aestronglyMeasurable] |
| simp only [q,cauchyObservation,cauchyMixture_at_own] |
| rw [heq] at hg |
| exact hg.trans (ownCauchyMixture_context_bound Γ κ he he1) |
| |
| |
| |
| def reciprocalScale {d : ℕ} (μ : Measure (Sphere d)) (s : Sphere d) : ℝ≥0∞ := |
| ∫⁻ t, (ENNReal.ofReal (dist s t))⁻¹ ∂μ |
| |
| def reciprocalCutoff (n : ℕ) : ℝ := 1/((n:ℝ)+1) |
| |
| lemma reciprocalCutoff_pos (n : ℕ) : 0 < reciprocalCutoff n := by unfold reciprocalCutoff; positivity |
| lemma reciprocalCutoff_le_one (n : ℕ) : reciprocalCutoff n ≤ 1 := by |
| unfold reciprocalCutoff |
| apply (div_le_iff₀ (by positivity)).mpr |
| simp |
| lemma reciprocalCutoff_antitone : Antitone reciprocalCutoff := by |
| intro n m h |
| unfold reciprocalCutoff |
| apply one_div_le_one_div_of_le (by positivity) |
| exact_mod_cast Nat.add_le_add_right h 1 |
| lemma reciprocalCutoff_limit : Tendsto reciprocalCutoff atTop (𝓝 0) := |
| tendsto_one_div_add_atTop_nhds_zero_nat |
| |
| lemma regularizedReciprocal_monotone {d : ℕ} (s t : Sphere d) : |
| Monotone (fun n => regularizedReciprocal (reciprocalCutoff n) s t) := by |
| intro n m h |
| unfold regularizedReciprocal |
| apply inv_anti₀ (lt_of_lt_of_le (reciprocalCutoff_pos m) (le_max_right _ _)) |
| exact max_le_max_left _ (reciprocalCutoff_antitone h) |
| |
| lemma regularizedReciprocal_limit {d : ℕ} (s t : Sphere d) : |
| Tendsto (fun n => ENNReal.ofReal (regularizedReciprocal (reciprocalCutoff n) s t)) atTop |
| (𝓝 (ENNReal.ofReal (dist s t))⁻¹) := by |
| have hm : Tendsto (fun n => ENNReal.ofReal (max (dist s t) (reciprocalCutoff n))) atTop |
| (𝓝 (ENNReal.ofReal (dist s t))) := by |
| simpa only [Function.comp_def,max_eq_left dist_nonneg] using |
| ENNReal.continuous_ofReal.continuousAt.tendsto.comp ((tendsto_const_nhds (x:=dist s t)).max reciprocalCutoff_limit) |
| have he : (fun n => ENNReal.ofReal (regularizedReciprocal (reciprocalCutoff n) s t)) = |
| fun n => (ENNReal.ofReal (max (dist s t) (reciprocalCutoff n)))⁻¹ := by |
| funext n |
| exact ENNReal.ofReal_inv_of_pos (lt_of_lt_of_le (reciprocalCutoff_pos n) (le_max_right _ _)) |
| rw [he] |
| exact tendsto_inv_iff.mpr hm |
| |
| lemma regularizedScale_ofReal {d : ℕ} (μ : Measure (Sphere d)) [IsProbabilityMeasure μ] |
| {e : ℝ} (he : 0 < e) (s : Sphere d) : |
| ENNReal.ofReal (regularizedScale μ e s) = ∫⁻ t, ENNReal.ofReal (regularizedReciprocal e s t) ∂μ := |
| ofReal_integral_eq_lintegral_ofReal (integrable_regularizedReciprocal μ he s) |
| (ae_of_all _ (fun t => (regularizedReciprocal_pos he s t).le)) |
| |
| lemma regularizedScale_limit {d : ℕ} (μ : Measure (Sphere d)) [IsProbabilityMeasure μ] (s : Sphere d) : |
| Tendsto (fun n => ENNReal.ofReal (regularizedScale μ (reciprocalCutoff n) s)) atTop |
| (𝓝 (reciprocalScale μ s)) := by |
| simp_rw [regularizedScale_ofReal μ (reciprocalCutoff_pos _)] |
| exact lintegral_tendsto_of_tendsto_of_monotone |
| (fun _ => (by unfold regularizedReciprocal; fun_prop)) |
| (ae_of_all _ (fun t => fun n m h => ENNReal.ofReal_le_ofReal (regularizedReciprocal_monotone s t h))) |
| (ae_of_all _ (fun t => regularizedReciprocal_limit s t)) |
| |
| lemma regularizedScale_monotone {d : ℕ} (μ : Measure (Sphere d)) [IsProbabilityMeasure μ] (s : Sphere d) : |
| Monotone (fun n => regularizedScale μ (reciprocalCutoff n) s) := by |
| intro n m h |
| exact integral_mono (integrable_regularizedReciprocal μ (reciprocalCutoff_pos n) s) |
| (integrable_regularizedReciprocal μ (reciprocalCutoff_pos m) s) |
| (fun t => regularizedReciprocal_monotone s t h) |
| |
| lemma reciprocalScale_kernel_measurable {d : ℕ} {C : Type*} [MeasurableSpace C] |
| (κ : Kernel C (Sphere d)) [IsMarkovKernel κ] : |
| Measurable (fun z : C×Sphere d => reciprocalScale (κ z.1) z.2) := by |
| exact (show Measurable (fun z : (C×Sphere d)×Sphere d => (ENNReal.ofReal (dist z.1.2 z.2))⁻¹) by |
| fun_prop).lintegral_kernel_prod_right (κ:=κ.comap Prod.fst measurable_fst) |
| |
| lemma regularizedScale_le_reciprocalScale {d : ℕ} (μ : Measure (Sphere d)) [IsProbabilityMeasure μ] |
| (s : Sphere d) (n : ℕ) : |
| ENNReal.ofReal (regularizedScale μ (reciprocalCutoff n) s) ≤ reciprocalScale μ s := by |
| exact ge_of_tendsto (regularizedScale_limit μ s) (eventually_atTop.mpr ⟨n,fun m hm => |
| ENNReal.ofReal_le_ofReal (regularizedScale_monotone μ s hm)⟩) |
| |
| lemma reciprocalScale_lower {d : ℕ} (μ : Measure (Sphere d)) [IsProbabilityMeasure μ] (s : Sphere d) : |
| ENNReal.ofReal (1/2:ℝ) ≤ reciprocalScale μ s := |
| (ENNReal.ofReal_le_ofReal (regularizedScale_bounds μ (reciprocalCutoff_pos 0) |
| (reciprocalCutoff_le_one 0) s).1).trans (regularizedScale_le_reciprocalScale μ s 0) |
| |
| def logScaleExtended (a : ℝ≥0∞) : ℝ≥0∞ := (ENNReal.log (2*a)).toENNReal |
| |
| lemma continuous_logScaleExtended : Continuous logScaleExtended := |
| EReal.continuous_toENNReal.comp (ENNReal.continuous_log.comp (ENNReal.continuous_const_mul (a:=2) (by norm_num))) |
| |
| lemma logScaleExtended_monotone : Monotone logScaleExtended := by |
| intro a b h |
| apply EReal.toENNReal_le_toENNReal |
| exact ENNReal.log_le_log (mul_le_mul_right h 2) |
| |
| lemma logScaleExtended_ofReal {a : ℝ} (ha : 0 < a) : |
| logScaleExtended (ENNReal.ofReal a) = ENNReal.ofReal (Real.log a+Real.log 2) := by |
| unfold logScaleExtended |
| rw [show (2:ℝ≥0∞)=ENNReal.ofReal (2:ℝ) by norm_num] |
| rw [←ENNReal.ofReal_mul (by norm_num : (0:ℝ)≤2),ENNReal.log_ofReal_of_pos (by positivity), |
| EReal.real_coe_toENNReal,Real.log_mul (by norm_num) ha.ne'] |
| congr 1 |
| ring |
| |
| lemma logScaleExtended_top_iff (a : ℝ≥0∞) : logScaleExtended a = ∞ ↔ a = ∞ := by |
| simp [logScaleExtended,ENNReal.mul_eq_top] |
| |
| lemma log_add_log_two_nonneg {a : ℝ} (ha : 1/2 ≤ a) : 0 ≤ Real.log a+Real.log 2 := by |
| have hp : 0 < a := lt_of_lt_of_le (by norm_num) ha |
| rw [←Real.log_mul hp.ne' (by norm_num)] |
| apply Real.log_nonneg |
| linarith |
| |
| lemma reciprocalScale_log_bound {d : ℕ} {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] |
| (κ : Kernel C (Sphere d)) [IsMarkovKernel κ] {M : ℝ} |
| (hb : ∀ n, (∫ z : C×Sphere d, Real.log (regularizedScale (κ z.1) (reciprocalCutoff n) z.2) ∂Γ) ≤ M) : |
| (∀ᵐ z : C×Sphere d ∂Γ, reciprocalScale (κ z.1) z.2 ≠ ∞) ∧ |
| Integrable (fun z : C×Sphere d => Real.log (reciprocalScale (κ z.1) z.2).toReal) Γ ∧ |
| (∫ z : C×Sphere d, Real.log (reciprocalScale (κ z.1) z.2).toReal ∂Γ) ≤ M := by |
| let f (n : ℕ) (z : C×Sphere d) := logScaleExtended (ENNReal.ofReal |
| (regularizedScale (κ z.1) (reciprocalCutoff n) z.2)) |
| let F (z : C×Sphere d) := logScaleExtended (reciprocalScale (κ z.1) z.2) |
| have hm n : Measurable (f n) := continuous_logScaleExtended.measurable.comp |
| (regularizedScale_kernel_measurable κ _).ennreal_ofReal |
| have hM : Measurable F := continuous_logScaleExtended.measurable.comp (reciprocalScale_kernel_measurable κ) |
| have hl (n : ℕ) (z : C×Sphere d) : 1/2 ≤ regularizedScale (κ z.1) (reciprocalCutoff n) z.2 := |
| (regularizedScale_bounds _ (reciprocalCutoff_pos n) (reciprocalCutoff_le_one n) _).1 |
| have he n : (∫⁻ z, f n z ∂Γ) = ENNReal.ofReal ((∫ z : C×Sphere d, |
| Real.log (regularizedScale (κ z.1) (reciprocalCutoff n) z.2) ∂Γ)+Real.log 2) := by |
| simp_rw [f,logScaleExtended_ofReal (lt_of_lt_of_le (by norm_num) (hl n _))] |
| rw [←ofReal_integral_eq_lintegral_ofReal (f:=fun z : C×Sphere d => Real.log (regularizedScale (κ z.1) (reciprocalCutoff n) z.2)+Real.log 2) |
| ((regularizedScale_context_log_integrable Γ κ (reciprocalCutoff_pos n) (reciprocalCutoff_le_one n)).add (integrable_const _)) |
| (ae_of_all _ (fun z => log_add_log_two_nonneg (hl n z))), |
| integral_add (regularizedScale_context_log_integrable Γ κ (reciprocalCutoff_pos n) |
| (reciprocalCutoff_le_one n)) (integrable_const _)] |
| simp |
| have hn : 0 ≤ M+Real.log 2 := by |
| have h := integral_nonneg (μ:=Γ) (f:=fun z : C×Sphere d => Real.log (regularizedScale (κ z.1) (reciprocalCutoff 0) z.2)+Real.log 2) (fun z => log_add_log_two_nonneg (hl 0 z)) |
| rw [integral_add (regularizedScale_context_log_integrable Γ κ (reciprocalCutoff_pos 0) |
| (reciprocalCutoff_le_one 0)) (integrable_const _)] at h |
| simp only [integral_const,probReal_univ,one_smul] at h |
| linarith [hb 0] |
| have ht : Tendsto (fun n => ∫⁻ z, f n z ∂Γ) atTop (𝓝 (∫⁻ z, F z ∂Γ)) := by |
| apply lintegral_tendsto_of_tendsto_of_monotone (fun n => (hm n).aemeasurable) |
| · exact ae_of_all _ (fun z n m h => logScaleExtended_monotone |
| (ENNReal.ofReal_le_ofReal (regularizedScale_monotone (κ z.1) z.2 h))) |
| · exact ae_of_all _ (fun z => continuous_logScaleExtended.continuousAt.tendsto.comp |
| (regularizedScale_limit (κ z.1) z.2)) |
| have hbF : (∫⁻ z, F z ∂Γ) ≤ ENNReal.ofReal (M+Real.log 2) := by |
| apply le_of_tendsto' ht |
| intro n |
| rw [he] |
| exact ENNReal.ofReal_le_ofReal (by linarith [hb n]) |
| have hfin : (∫⁻ z, F z ∂Γ) ≠ ∞ := ne_top_of_le_ne_top ENNReal.ofReal_ne_top hbF |
| have haeF : ∀ᵐ z ∂Γ, F z ≠ ∞ := (ae_lt_top hM hfin).mono (fun _ h => h.ne) |
| have hae : ∀ᵐ z : C×Sphere d ∂Γ, reciprocalScale (κ z.1) z.2 ≠ ∞ := by |
| filter_upwards [haeF] with z hz |
| exact mt (logScaleExtended_top_iff _).mpr hz |
| have heF : (fun z => (F z).toReal) =ᵐ[Γ] |
| fun z : C×Sphere d => Real.log (reciprocalScale (κ z.1) z.2).toReal+Real.log 2 := by |
| filter_upwards [hae] with z hz |
| have hlz : 1/2 ≤ (reciprocalScale (κ z.1) z.2).toReal := by |
| exact (ENNReal.ofReal_le_iff_le_toReal hz).mp (reciprocalScale_lower _ _) |
| have hez := logScaleExtended_ofReal (lt_of_lt_of_le (by norm_num) hlz) |
| rw [ENNReal.ofReal_toReal hz] at hez |
| change (logScaleExtended _).toReal = _ |
| rw [hez,ENNReal.toReal_ofReal (log_add_log_two_nonneg hlz)] |
| have hi := integrable_toReal_of_lintegral_ne_top hM.aemeasurable hfin |
| have hi' : Integrable (fun z : C×Sphere d => Real.log (reciprocalScale (κ z.1) z.2).toReal) Γ := by |
| have := (hi.congr heF).sub (integrable_const (Real.log 2)) |
| convert this using 1 |
| ext z |
| simp |
| refine ⟨hae,hi',?_⟩ |
| have hreal := ENNReal.toReal_mono ENNReal.ofReal_ne_top hbF |
| rw [ENNReal.toReal_ofReal hn,←integral_toReal hM.aemeasurable (haeF.mono (fun _ h => h.lt_top)), |
| integral_congr_ae heF,integral_add hi' (integrable_const _)] at hreal |
| simp only [integral_const,probReal_univ,one_smul] at hreal |
| linarith |
| |
| lemma reciprocalScale_experiment_bound {d : ℕ} {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] |
| (κ : Kernel C (Sphere d)) [IsMarkovKernel κ] |
| (hfin : conditionalReferenceKL (cauchyObservationLaw Γ) (gaussianReal 0 1) ≠ ∞) : |
| (∀ᵐ z : C×Sphere d ∂Γ, reciprocalScale (κ z.1) z.2 ≠ ∞) ∧ |
| Integrable (fun z : C×Sphere d => Real.log (reciprocalScale (κ z.1) z.2).toReal) Γ ∧ |
| (∫ z : C×Sphere d, Real.log (reciprocalScale (κ z.1) z.2).toReal ∂Γ) ≤ |
| cauchyTiltConstant-standardGaussianEntropy+ |
| (conditionalReferenceKL (cauchyObservationLaw Γ) (gaussianReal 0 1)).toReal := by |
| apply reciprocalScale_log_bound Γ κ |
| intro n |
| have h := cauchyMixture_experiment_bound Γ κ hfin (reciprocalCutoff_pos n) (reciprocalCutoff_le_one n) |
| linarith |
| |
| |
| |
| def residualObservation {d r : ℕ} {C : Type*} (z : (C×Sphere d)×Rows r d) : |
| (C×(Rows r d×(Fin r→ℝ)))×Sphere d := |
| ((z.1.1,(z.2,label z.2 (z.1.2:Vector d))),z.1.2) |
| |
| lemma measurable_residualObservation {d r : ℕ} {C : Type*} [MeasurableSpace C] : |
| Measurable (residualObservation (d:=d) (r:=r) (C:=C)) := by |
| unfold residualObservation label |
| fun_prop |
| |
| def residualLaw {d : ℕ} {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) (r : ℕ) : Measure ((C×(Rows r d×(Fin r→ℝ)))×Sphere d) := |
| (Γ.prod (gaussianRows r d)).map residualObservation |
| |
| instance residualLaw_probability {d : ℕ} {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] (r : ℕ) : |
| IsProbabilityMeasure (residualLaw Γ r) := by |
| exact (Measure.isProbabilityMeasure_map_iff |
| measurable_residualObservation.aemeasurable).mpr inferInstance |
| |
| lemma residualLaw_support {d r : ℕ} {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] : |
| ∀ᵐ z ∂residualLaw Γ r, label z.1.2.1 (z.2:Vector d) = z.1.2.2 := by |
| apply (ae_map_iff measurable_residualObservation.aemeasurable |
| (measurableSet_eq_fun (by unfold label; fun_prop) (by fun_prop))).mpr |
| exact ae_of_all _ (fun _ => rfl) |
| |
| lemma residualPosterior_support {d r : ℕ} [Nonempty (Sphere d)] {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] : |
| ∀ᵐ c ∂(residualLaw Γ r).fst, ∀ᵐ t : Sphere d ∂(residualLaw Γ r).condKernel c, |
| label c.2.1 (t:Vector d) = c.2.2 := by |
| apply Measure.ae_ae_of_ae_compProd (p:=fun z : (C×(Rows r d×(Fin r→ℝ)))×Sphere d => label z.1.2.1 (z.2:Vector d) = z.1.2.2) |
| rw [(residualLaw Γ r).disintegrate (residualLaw Γ r).condKernel] |
| exact residualLaw_support Γ |
| |
| lemma reciprocalScale_finite_nonatomicAt {d : ℕ} (μ : Measure (Sphere d)) (s : Sphere d) |
| (h : reciprocalScale μ s ≠ ∞) : ∀ᵐ t ∂μ, t ≠ s := by |
| have ha := ae_lt_top (show Measurable (fun t : Sphere d => (ENNReal.ofReal (dist s t))⁻¹) by fun_prop) h |
| filter_upwards [ha] with t ht |
| rintro rfl |
| simp at ht |
| |
| lemma reciprocalScale_finite_eq_real {d : ℕ} (μ : Measure (Sphere d)) (s : Sphere d) |
| (h : reciprocalScale μ s ≠ ∞) : |
| reciprocalScale μ s = ∫⁻ t, ENNReal.ofReal (‖(t:Vector d)-(s:Vector d)‖⁻¹) ∂μ := by |
| apply lintegral_congr_ae |
| filter_upwards [reciprocalScale_finite_nonatomicAt μ s h] with t ht |
| have hn : 0 < ‖(t:Vector d)-(s:Vector d)‖ := norm_pos_iff.mpr |
| (sub_ne_zero.mpr (fun he => ht (Subtype.ext he))) |
| rw [ENNReal.ofReal_inv_of_pos hn] |
| congr 2 |
| change dist (s:Vector d) (t:Vector d) = _ |
| rw [dist_comm,dist_eq_norm] |
| |
| lemma label_eq_difference_orthogonal {d r : ℕ} (A : Rows r d) (s t : Vector d) |
| (h : label A t = label A s) : t-s ∈ (rowSpan A)ᗮ := by |
| apply (Submodule.mem_orthogonal' _ _).mpr |
| intro u hu |
| induction hu using Submodule.span_induction with |
| | mem v hv => |
| obtain ⟨i,rfl⟩ := hv |
| rw [inner_sub_left,sub_eq_zero] |
| exact congrFun h i |
| | zero => simp |
| | add x y hx hy hx' hy' => simp [inner_add_right,hx',hy'] |
| | smul a x hx hx' => simp [inner_smul_right,hx'] |
| |
| |
| |
| def fixedTargetPosterior {d r : ℕ} [Nonempty (Sphere d)] {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] (c : C) (s : Sphere d) : |
| Kernel (Rows r d) (Vector d) := |
| ((residualLaw Γ r).condKernel.comap (fun A => (c,(A,label A (s:Vector d)))) |
| (by unfold label; fun_prop)).map Subtype.val |
| |
| instance fixedTargetPosterior_markov {d r : ℕ} [Nonempty (Sphere d)] {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] (c : C) (s : Sphere d) : |
| IsMarkovKernel (fixedTargetPosterior (r:=r) Γ c s) := by |
| unfold fixedTargetPosterior |
| exact Kernel.IsMarkovKernel.map _ measurable_subtype_coe |
| |
| lemma fixedTargetPosterior_support {d r : ℕ} [Nonempty (Sphere d)] {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] |
| (hfin : ∀ᵐ z ∂residualLaw Γ r, reciprocalScale ((residualLaw Γ r).condKernel z.1) z.2 ≠ ∞) : |
| ∀ᵐ z : C×Sphere d ∂Γ, ∀ᵐ w ∂(gaussianRows r d) ⊗ₘ fixedTargetPosterior Γ z.1 z.2, |
| w.2-(z.2:Vector d) ≠ 0 ∧ w.2-(z.2:Vector d) ∈ (rowSpan w.1)ᗮ := by |
| have hs := ae_of_ae_map measurable_residualObservation.aemeasurable |
| (ae_of_ae_map measurable_fst.aemeasurable (residualPosterior_support Γ (r:=r))) |
| have hf := ae_of_ae_map measurable_residualObservation.aemeasurable hfin |
| filter_upwards [Measure.ae_ae_of_ae_prod hs,Measure.ae_ae_of_ae_prod hf] with z hzs hzf |
| apply Measure.ae_compProd_of_ae_ae |
| · have hm : Measurable (fun w : Rows r d×Vector d => label w.1 (w.2-(z.2:Vector d))) := by |
| unfold label |
| fun_prop |
| have he : {w : Rows r d×Vector d | w.2-(z.2:Vector d) ≠ 0 ∧ |
| w.2-(z.2:Vector d) ∈ (rowSpan w.1)ᗮ} = |
| {w | w.2-(z.2:Vector d) ≠ 0 ∧ label w.1 (w.2-(z.2:Vector d)) = 0} := by |
| ext w |
| simp only [Set.mem_ofPred_eq,and_congr_right_iff] |
| intro _ |
| constructor |
| · intro h |
| funext i |
| exact Submodule.inner_left_of_mem_orthogonal |
| (show w.1 i ∈ rowSpan w.1 from Submodule.subset_span ⟨i,rfl⟩) h |
| · intro h |
| have he : label w.1 w.2 = label w.1 (z.2:Vector d) := by |
| funext i |
| have hi := congrFun h i |
| simp only [label,inner_sub_left,Pi.zero_apply] at hi |
| exact sub_eq_zero.mp hi |
| exact label_eq_difference_orthogonal _ _ _ he |
| rw [he] |
| exact (measurableSet_eq_fun (by fun_prop) measurable_const).compl.inter |
| (measurableSet_eq_fun hm measurable_const) |
| · filter_upwards [hzs,hzf] with A hAs hAf |
| unfold fixedTargetPosterior |
| rw [Kernel.map_apply _ measurable_subtype_coe] |
| change ∀ᵐ t ∂(((residualLaw Γ r).condKernel (z.1,(A,label A (z.2:Vector d)))).map Subtype.val), |
| t-(z.2:Vector d) ≠ 0 ∧ t-(z.2:Vector d) ∈ (rowSpan A)ᗮ |
| apply (ae_map_iff measurable_subtype_coe.aemeasurable |
| ((measurableSet_eq_fun (by fun_prop) measurable_const).compl.inter |
| ((Submodule.isClosed_orthogonal (rowSpan A)).measurableSet.preimage (by fun_prop)))).mpr |
| filter_upwards [hAs,reciprocalScale_finite_nonatomicAt _ _ hAf] with t ht hne |
| exact ⟨sub_ne_zero.mpr (fun h => hne (Subtype.ext h)),label_eq_difference_orthogonal A _ _ ht⟩ |
| |
| |
| |
| lemma mixed_entropy_log_bound_ae {A X Z : Type*} [MeasurableSpace A] [MeasurableSpace X] |
| [MeasurableSpace Z] (P : Measure (A×X)) [IsProbabilityMeasure P] |
| (γ : Measure X) (η : Measure Z) [IsProbabilityMeasure γ] [IsProbabilityMeasure η] |
| (hKL : klDiv P (P.fst.prod γ) ≠ ∞) |
| (K : (A×X)×Z → ℝ≥0∞) (hKm : Measurable K) (hKt : ∀ z, K z ≠ ∞) |
| (hKint : ∀ ax, ∫⁻ z, K (ax,z) ∂η ≠ ∞) |
| (C : A → ℝ) (hCm : Measurable C) (hC0 : ∀ a, 0 ≤ C a) |
| (hCI : Integrable (fun a => Real.log (1+C a)) P.fst) |
| {q : ℕ} (hq : 1 ≤ q) |
| (hMoment : ∀ᵐ a ∂P.fst, ∫⁻ x, (∫⁻ z, K ((a,x),z) ∂η)^q ∂γ ≤ (ENNReal.ofReal (C a))^q) : |
| ∫⁻ axz, ENNReal.ofReal (Real.log (1+(K axz).toReal)) ∂P.prod η ≤ |
| ENNReal.ofReal ((klDiv P (P.fst.prod γ)).toReal / q + |
| ∫ a, Real.log (1+C a) ∂P.fst) := by |
| let F : A×X → ℝ≥0∞ := fun ax => ∫⁻ z, K (ax,z) ∂η |
| have hFm : Measurable F := Measurable.lintegral_prod_right (f:=fun ax z => K (ax,z)) hKm |
| have hf : Measurable (fun ax : A×X => (q:ℝ)*Real.log (1+(F ax).toReal)) := |
| measurable_const.mul (measurable_log_one_add hFm) |
| have hc : Measurable (fun ax : A×X => (q:ℝ)*Real.log (1+C ax.1)) := |
| measurable_const.mul ((measurable_const.add (hCm.comp measurable_fst)).log) |
| have hCc : Integrable (fun ax : A×X => Real.log (1+C ax.1)) P := |
| (integrable_map_measure (measurable_const.add hCm |>.log.aestronglyMeasurable) |
| measurable_fst.aemeasurable).mp hCI |
| have hcI := hCc.const_mul (q:ℝ) |
| have hExp : ∫⁻ ax : A×X, ENNReal.ofReal (Real.exp |
| ((q:ℝ)*Real.log (1+(F ax).toReal)-(q:ℝ)*Real.log (1+C ax.1))) ∂P.fst.prod γ ≤ 1 := by |
| rw [lintegral_prod (fun ax : A×X => ENNReal.ofReal (Real.exp |
| ((q:ℝ)*Real.log (1+(F ax).toReal)-(q:ℝ)*Real.log (1+C ax.1)))) |
| (by exact (hf.sub hc).exp.ennreal_ofReal.aemeasurable)] |
| calc |
| _ ≤ ∫⁻ _a, (1:ℝ≥0∞) ∂P.fst := by |
| apply lintegral_mono_ae |
| filter_upwards [hMoment] with a hMa |
| have heq : (fun x => ENNReal.ofReal (Real.exp |
| ((q:ℝ)*Real.log (1+(F (a,x)).toReal)-(q:ℝ)*Real.log (1+C a)))) = |
| (fun x => (1+F (a,x))^q / (ENNReal.ofReal (1+C a))^q) := by |
| funext x |
| exact exp_log_normalization _ (hKint (a,x)) _ (hC0 a) q |
| change (∫⁻ x, ENNReal.ofReal (Real.exp |
| ((q:ℝ)*Real.log (1+(F (a,x)).toReal)-(q:ℝ)*Real.log (1+C a))) ∂γ) ≤ 1 |
| rw [heq] |
| simp_rw [div_eq_mul_inv] |
| have hpow : Measurable (fun x : X => (1+F (a,x))^q) := |
| (measurable_const.add (hFm.comp (measurable_const.prodMk measurable_id))).pow_const q |
| rw [lintegral_mul_const _ hpow] |
| have hh := lintegral_one_add_pow_le γ (fun x => F (a,x)) |
| (hFm.comp (measurable_const.prodMk measurable_id)) hq hMa |
| have he : ENNReal.ofReal (1+C a) = 1+ENNReal.ofReal (C a) := by |
| rw [ENNReal.ofReal_add (by norm_num) (hC0 a), ENNReal.ofReal_one] |
| rw [he] |
| apply (mul_le_mul' hh le_rfl).trans_eq |
| exact ENNReal.mul_inv_cancel (by positivity) (by finiteness) |
| _ = 1 := by simp |
| obtain ⟨hfi, hbi⟩ := entropy_exp_bound P (P.fst.prod γ) hKL _ _ hf hc |
| (fun ax => mul_nonneg (Nat.cast_nonneg _) (log_one_add_nonneg _)) hcI hExp |
| have hqpos : 0 < (q:ℝ) := by exact_mod_cast (show 0 < q by omega) |
| have hqne := hqpos.ne' |
| have hLi : Integrable (fun ax => Real.log (1+(F ax).toReal)) P := by |
| have hh := hfi.div_const (q:ℝ) |
| simpa only [mul_div_cancel_left₀ _ hqne] using hh |
| have hCeq : ∫ ax : A×X, Real.log (1+C ax.1) ∂P = ∫ a, Real.log (1+C a) ∂P.fst := by |
| exact (integral_map measurable_fst.aemeasurable |
| (measurable_const.add hCm |>.log.aestronglyMeasurable)).symm |
| rw [integral_const_mul, integral_const_mul, hCeq] at hbi |
| have hfinal : ∫ ax, Real.log (1+(F ax).toReal) ∂P ≤ |
| (klDiv P (P.fst.prod γ)).toReal / q + ∫ a, Real.log (1+C a) ∂P.fst := by |
| rw [← sub_le_iff_le_add, le_div_iff₀ hqpos] |
| nlinarith |
| calc |
| _ = ∫⁻ ax, ∫⁻ z, ENNReal.ofReal (Real.log (1+(K (ax,z)).toReal)) ∂η ∂P := |
| lintegral_prod _ (measurable_log_one_add hKm |>.ennreal_ofReal.aemeasurable) |
| _ ≤ ∫⁻ ax, ENNReal.ofReal (Real.log (1+(F ax).toReal)) ∂P := by |
| exact lintegral_mono fun ax => log_one_add_jensen η (fun z => K (ax,z)) |
| (hKm.comp (measurable_const.prodMk measurable_id)) (fun z => hKt (ax,z)) (hKint ax) |
| _ = ENNReal.ofReal (∫ ax, Real.log (1+(F ax).toReal) ∂P) := |
| (ofReal_integral_eq_lintegral_ofReal hLi (Filter.Eventually.of_forall fun _ => log_one_add_nonneg _)).symm |
| _ ≤ _ := ENNReal.ofReal_le_ofReal hfinal |
| |
| |
| |
| def residualScale {d r : ℕ} [Nonempty (Sphere d)] {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] |
| (z : C×Sphere d) (A : Rows r d) : ℝ≥0∞ := |
| reciprocalScale ((residualLaw Γ r).condKernel (z.1,(A,label A (z.2:Vector d)))) z.2 |
| |
| lemma measurable_residualScale {d r : ℕ} [Nonempty (Sphere d)] {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] : |
| Measurable (fun z : (C×Sphere d)×Rows r d => residualScale Γ z.1 z.2) := by |
| exact (reciprocalScale_kernel_measurable (residualLaw Γ r).condKernel).comp |
| measurable_residualObservation |
| |
| lemma residualScale_lower {d r : ℕ} [Nonempty (Sphere d)] {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] (z : C×Sphere d) (A : Rows r d) : |
| ENNReal.ofReal (1/2:ℝ) ≤ residualScale Γ z A := reciprocalScale_lower _ _ |
| |
| lemma fixedTargetPosterior_scale {d r : ℕ} [Nonempty (Sphere d)] {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] (z : C×Sphere d) (A : Rows r d) |
| (hf : residualScale Γ z A ≠ ∞) : |
| reciprocalKernelScale (fixedTargetPosterior Γ z.1 z.2) (z.2:Vector d) A = residualScale Γ z A := by |
| unfold reciprocalKernelScale fixedTargetPosterior |
| rw [Kernel.map_apply _ measurable_subtype_coe] |
| rw [lintegral_map (by fun_prop) measurable_subtype_coe] |
| exact (reciprocalScale_finite_eq_real _ _ hf).symm |
| |
| def residualCube {d r : ℕ} [Nonempty (Sphere d)] {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] (δ : ℝ) |
| (w : ((C×Sphere d)×Vector d)×Rows r d) : ℝ≥0∞ := |
| (∫⁻ t : Sphere d, cubeKernel δ ⟪(t:Vector d)-(w.1.1.2:Vector d),w.1.2⟫ |
| ∂(residualLaw Γ r).condKernel (w.1.1.1,(w.2,label w.2 (w.1.1.2:Vector d)))) / |
| residualScale Γ w.1.1 w.2 |
| |
| lemma measurable_residualCube {d r : ℕ} [Nonempty (Sphere d)] {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] (δ : ℝ) : |
| Measurable (residualCube (r:=r) Γ δ) := by |
| let f : ((C×Sphere d)×Vector d)×Rows r d → C×(Rows r d×(Fin r→ℝ)) := |
| fun w => (w.1.1.1,(w.2,label w.2 (w.1.1.2:Vector d))) |
| have hf : Measurable f := by unfold f label; fun_prop |
| have hn : Measurable (fun w : ((C×Sphere d)×Vector d)×Rows r d => |
| ∫⁻ t : Sphere d, cubeKernel δ ⟪(t:Vector d)-(w.1.1.2:Vector d),w.1.2⟫ |
| ∂(residualLaw Γ r).condKernel (f w)) := |
| (show Measurable (fun z : (((C×Sphere d)×Vector d)×Rows r d)×Sphere d => |
| cubeKernel δ ⟪(z.2:Vector d)-(z.1.1.1.2:Vector d),z.1.1.2⟫) from |
| (measurable_cubeKernel δ).comp (by fun_prop)).lintegral_kernel_prod_right |
| (f:=fun w t => cubeKernel δ ⟪(t:Vector d)-(w.1.1.2:Vector d),w.1.2⟫) |
| (κ:=(residualLaw Γ r).condKernel.comap f hf) |
| exact hn.div ((measurable_residualScale Γ).comp |
| (measurable_fst.fst.prodMk measurable_snd)) |
| |
| lemma residualCube_bound {d r : ℕ} [Nonempty (Sphere d)] {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] (δ : ℝ) |
| (w : ((C×Sphere d)×Vector d)×Rows r d) : |
| residualCube Γ δ w ≤ ENNReal.ofReal ((2*δ)⁻¹) * 2 := by |
| have hn : (∫⁻ t : Sphere d, cubeKernel δ ⟪(t:Vector d)-(w.1.1.2:Vector d),w.1.2⟫ |
| ∂(residualLaw Γ r).condKernel (w.1.1.1,(w.2,label w.2 (w.1.1.2:Vector d)))) ≤ |
| ENNReal.ofReal ((2*δ)⁻¹) := by |
| calc |
| _ ≤ ∫⁻ _t : Sphere d, ENNReal.ofReal ((2*δ)⁻¹) |
| ∂(residualLaw Γ r).condKernel (w.1.1.1,(w.2,label w.2 (w.1.1.2:Vector d))) := |
| lintegral_mono (fun t => by |
| unfold cubeKernel |
| exact Set.indicator_le_self _ _ _) |
| _ = _ := by simp |
| have ha : (residualScale Γ w.1.1 w.2)⁻¹ ≤ (2:ℝ≥0∞) := by |
| have h := ENNReal.inv_le_inv.mpr (residualScale_lower Γ w.1.1 w.2) |
| have he : (ENNReal.ofReal (1/2:ℝ))⁻¹ = (2:ℝ≥0∞) := by |
| rw [ENNReal.ofReal_div_of_pos (by norm_num),ENNReal.ofReal_one,ENNReal.ofReal_ofNat] |
| simp |
| rwa [he] at h |
| exact mul_le_mul' hn ha |
| |
| lemma residualCube_finite {d r : ℕ} [Nonempty (Sphere d)] {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] (δ : ℝ) |
| (w : ((C×Sphere d)×Vector d)×Rows r d) : residualCube Γ δ w ≠ ∞ := |
| ne_top_of_le_ne_top (by finiteness) (residualCube_bound Γ δ w) |
| |
| lemma residualCube_integral_finite {d r : ℕ} [Nonempty (Sphere d)] {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] (δ : ℝ) |
| (w : (C×Sphere d)×Vector d) : |
| (∫⁻ A, residualCube Γ δ (w,A) ∂gaussianRows r d) ≠ ∞ := by |
| apply ne_top_of_le_ne_top (b:=ENNReal.ofReal ((2*δ)⁻¹)*2) (by finiteness) |
| exact (lintegral_mono (fun A => residualCube_bound Γ δ (w,A))).trans_eq (by simp) |
| |
| lemma residualCube_moment {d r q : ℕ} [Nonempty (Sphere d)] {C : Type*} [MeasurableSpace C] |
| (Γ : Measure (C×Sphere d)) [IsProbabilityMeasure Γ] |
| (hr : r ≤ d) (hd : d ≤ 8*r) (hq : 2*q ≤ r) (hmargin : 2 ≤ r-q) |
| (hfin : ∀ᵐ z ∂residualLaw Γ r, reciprocalScale ((residualLaw Γ r).condKernel z.1) z.2 ≠ ∞) |
| {δ : ℝ} (hδ : 0 < δ) : |
| ∀ᵐ z : C×Sphere d ∂Γ, ∫⁻ x, (∫⁻ A, residualCube Γ δ ((z,x),A) |
| ∂gaussianRows r d)^q ∂stdGaussian (Vector d) ≤ ENNReal.ofReal (5*kernelSmallBallConstant)^q := by |
| have hdpos : 0 < d := by omega |
| let : Nonempty (Fin d) := Fin.pos_iff_nonempty.mp hdpos |
| have hf := Measure.ae_ae_of_ae_prod (ae_of_ae_map measurable_residualObservation.aemeasurable hfin) |
| filter_upwards [fixedTargetPosterior_support Γ hfin,hf] with z hzs hzf |
| have heq : ∀ᵐ A ∂gaussianRows r d, |
| reciprocalKernelScale (fixedTargetPosterior Γ z.1 z.2) (z.2:Vector d) A = residualScale Γ z A := by |
| filter_upwards [hzf] with A hA |
| exact fixedTargetPosterior_scale Γ z A hA |
| have hscale : ∀ᵐ A ∂gaussianRows r d, |
| reciprocalKernelScale (fixedTargetPosterior Γ z.1 z.2) (z.2:Vector d) A ≠ 0 ∧ |
| reciprocalKernelScale (fixedTargetPosterior Γ z.1 z.2) (z.2:Vector d) A ≠ ∞ := by |
| filter_upwards [heq,hzf] with A he hf |
| rw [he] |
| exact ⟨ne_of_gt (lt_of_lt_of_le (by norm_num) (residualScale_lower Γ z A)),hf⟩ |
| have h := reciprocal_kernel_projection_moment (E:=Vector d) |
| (by simpa) (by simpa) hq hmargin (fixedTargetPosterior Γ z.1 z.2) (z.2:Vector d) |
| hscale hzs (cubeKernel δ) (measurable_cubeKernel δ) (lintegral_cubeKernel hδ) |
| apply le_trans (le_of_eq ?_) h |
| apply lintegral_congr |
| intro x |
| congr 1 |
| apply lintegral_congr_ae |
| filter_upwards [heq] with A he |
| rw [he] |
| unfold residualCube fixedTargetPosterior |
| rw [Kernel.map_apply _ measurable_subtype_coe,lintegral_map |
| (f:=fun t : Vector d => cubeKernel δ ⟪t-(z.2:Vector d),x⟫) |
| ((measurable_cubeKernel δ).comp (by fun_prop)) measurable_subtype_coe] |
| rfl |
| |
| |
| |
| |
| def scalarCube (μ : Measure ℝ) (y δ : ℝ) : ℝ≥0∞ := |
| ∫⁻ t, cubeKernel δ (t-y) ∂μ |
| |
| lemma scalarCube_ball (μ : Measure ℝ) (y : ℝ) {δ : ℝ} (hδ : 0 < δ) : |
| scalarCube μ y δ = μ (Metric.closedBall y δ) / volume (Metric.closedBall y δ) := by |
| have he : (fun t : ℝ => cubeKernel δ (t-y)) = |
| Set.indicator (Metric.closedBall y δ) (fun _ => ENNReal.ofReal ((2*δ)⁻¹)) := by |
| ext t |
| simp only [cubeKernel,Set.indicator_apply] |
| congr 1 |
| simp only [Set.mem_Icc,Metric.mem_closedBall,Real.dist_eq,abs_le] |
| unfold scalarCube |
| rw [he,lintegral_indicator Metric.isClosed_closedBall.measurableSet, |
| lintegral_const,Measure.restrict_apply_univ,Real.volume_closedBall, |
| ENNReal.ofReal_inv_of_pos (by positivity),div_eq_mul_inv,mul_comm] |
| |
| lemma measurable_scalarCube {A : Type*} [MeasurableSpace A] |
| (κ : Kernel A ℝ) [IsSFiniteKernel κ] (δ : ℝ) : |
| Measurable (fun z : A×ℝ => scalarCube (κ z.1) z.2 δ) := by |
| exact ((measurable_cubeKernel δ).comp (show Measurable (fun z : (A×ℝ)×ℝ => z.2-z.1.2) by fun_prop)).lintegral_kernel_prod_right |
| (f:=fun z t => cubeKernel δ (t-z.2)) (κ:=κ.comap Prod.fst measurable_fst) |
| |
| def scalarExtended (μ : Measure ℝ) (y : ℝ) : ℝ≥0∞ := |
| liminf (fun n : ℕ => scalarCube μ y (1/(n:ℝ))) atTop |
| |
| def scalarDensity (μ : Measure ℝ) (y : ℝ) : ℝ≥0∞ := |
| if scalarExtended μ y = ∞ then 0 else scalarExtended μ y |
| |
| lemma measurable_scalarExtended {A : Type*} [MeasurableSpace A] |
| (κ : Kernel A ℝ) [IsSFiniteKernel κ] : |
| Measurable (fun z : A×ℝ => scalarExtended (κ z.1) z.2) := |
| Measurable.liminf (fun n => measurable_scalarCube κ (1/(n:ℝ))) |
| |
| lemma measurable_scalarDensity {A : Type*} [MeasurableSpace A] |
| (κ : Kernel A ℝ) [IsSFiniteKernel κ] : |
| Measurable (fun z : A×ℝ => scalarDensity (κ z.1) z.2) := by |
| exact Measurable.ite (measurableSet_eq_fun (measurable_scalarExtended κ) measurable_const) |
| measurable_const (measurable_scalarExtended κ) |
| |
| lemma scalarDensity_finite (μ : Measure ℝ) (y : ℝ) : scalarDensity μ y ≠ ∞ := by |
| unfold scalarDensity |
| split_ifs with h <;> simp_all |
| |
| lemma scalarDensity_ae_rnDeriv (μ : Measure ℝ) [IsFiniteMeasure μ] : |
| scalarDensity μ =ᵐ[volume] μ.rnDeriv volume := by |
| filter_upwards [Besicovitch.ae_tendsto_rnDeriv μ volume,Measure.rnDeriv_ne_top μ volume] with y hy hyt |
| have ht := hy.comp tendsto_cube_scale |
| have he : (fun n : ℕ => scalarCube μ y (1/(n:ℝ))) =ᶠ[atTop] |
| fun n : ℕ => μ (Metric.closedBall y (1/(n:ℝ))) / volume (Metric.closedBall y (1/(n:ℝ))) := by |
| filter_upwards [eventually_ge_atTop 1] with n hn |
| exact scalarCube_ball μ y (by positivity) |
| have hl : scalarExtended μ y = μ.rnDeriv volume y := (ht.congr' he.symm).liminf_eq |
| simp only [scalarDensity,hl,ite_eq_right hyt] |
| |
| lemma scalarDensity_ae_limit (μ : Measure ℝ) [IsFiniteMeasure μ] : |
| ∀ᵐ y ∂(volume : Measure ℝ), Tendsto (fun n : ℕ => scalarCube μ y (1/(n:ℝ))) |
| atTop (𝓝 (scalarDensity μ y)) := by |
| filter_upwards [Besicovitch.ae_tendsto_rnDeriv μ volume,scalarDensity_ae_rnDeriv μ] with y hy heq |
| have he : (fun n : ℕ => scalarCube μ y (1/(n:ℝ))) =ᶠ[atTop] |
| fun n : ℕ => μ (Metric.closedBall y (1/(n:ℝ))) / volume (Metric.closedBall y (1/(n:ℝ))) := by |
| filter_upwards [eventually_ge_atTop 1] with n hn |
| exact scalarCube_ball μ y (by positivity) |
| rw [heq] |
| exact (hy.comp tendsto_cube_scale).congr' he.symm |
| |
| lemma scalarDensity_withDensity (μ : Measure ℝ) [IsFiniteMeasure μ] (h : μ ≪ volume) : |
| volume.withDensity (scalarDensity μ) = μ := by |
| rw [withDensity_congr_ae (scalarDensity_ae_rnDeriv μ)] |
| exact Measure.withDensity_rnDeriv_eq _ _ h |
| |
| lemma kernel_scalarDensity_withDensity {A : Type*} [MeasurableSpace A] |
| (π : Measure A) [SFinite π] (κ : Kernel A ℝ) [IsFiniteKernel κ] |
| (h : ∀ᵐ a ∂π, κ a ≪ volume) : |
| (π.prod volume).withDensity (fun z : A×ℝ => scalarDensity (κ z.1) z.2) = π ⊗ₘ κ := by |
| apply Measure.ext |
| intro t ht |
| rw [withDensity_apply _ ht,←lintegral_indicator ht, |
| lintegral_prod _ ((measurable_scalarDensity κ).indicator ht).aemeasurable, |
| Measure.compProd_apply ht] |
| apply lintegral_congr_ae |
| filter_upwards [h] with a ha |
| have ht' : MeasurableSet (Prod.mk a ⁻¹' t) := ht.preimage (measurable_const.prodMk measurable_id) |
| rw [←scalarDensity_withDensity (κ a) ha,withDensity_apply _ ht',←lintegral_indicator ht'] |
| rfl |
| |
| lemma kernel_scalarDensity_ae_limit {A : Type*} [MeasurableSpace A] |
| (π : Measure A) [SFinite π] (κ : Kernel A ℝ) [IsFiniteKernel κ] : |
| ∀ᵐ z : A×ℝ ∂π.prod volume, Tendsto |
| (fun n : ℕ => scalarCube (κ z.1) z.2 (1/(n:ℝ))) atTop (𝓝 (scalarDensity (κ z.1) z.2)) := by |
| have hm : MeasurableSet {z : A×ℝ | Tendsto |
| (fun n : ℕ => scalarCube (κ z.1) z.2 (1/(n:ℝ))) atTop (𝓝 (scalarDensity (κ z.1) z.2))} := |
| measurableSet_tendsto_fun (fun n : ℕ => measurable_scalarCube κ (1/(n:ℝ))) (measurable_scalarDensity κ) |
| exact (Measure.ae_prod_iff_ae_ae hm).mpr (ae_of_all _ (fun a => scalarDensity_ae_limit (κ a))) |
| |
| |
| |
| end CurrentProjection |
| |
| end |
| |
| end OAI |
| |
