| import OAI.Probability.PosteriorReplicas.Geometry.LabelEllipsoids |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set Matrix InnerProductSpace Filter Metric |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| noncomputable section |
| namespace PosteriorReplicas |
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| lemma liftLabel_neg {d k : ℕ} (A : Rows d k) (y : Labels k) : |
| liftLabel A (-y) = -liftLabel A y := by |
| simp [liftLabel, Matrix.mulVec_neg, Finset.sum_neg_distrib] |
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| lemma norm_liftLabel_sub_comm {d k : ℕ} (A : Rows d k) (x y : Labels k) : |
| ‖liftLabel A (x-y)‖ = ‖liftLabel A (y-x)‖ := by |
| rw [← neg_sub y x, liftLabel_neg, norm_neg] |
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| def ellipsoidMass {d k : ℕ} (ν : Measure (Vec d)) (A : Rows d k) (y : Labels k) (r : ℝ) : ℝ≥0∞ := |
| ν {x | ‖liftLabel A (labels A x-y)‖ ≤ r} |
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| @[fun_prop] lemma measurable_ellipsoidMass {d k : ℕ} (ν : Measure (Vec d)) [SFinite ν] (r : ℝ) : |
| Measurable (fun z : Data d k => ellipsoidMass ν z.1 z.2 r) := by |
| change Measurable (fun z : Data d k => ν ((Prod.mk z) ⁻¹' |
| {q : Data d k × Vec d | ‖liftLabel q.1.1 (labels q.1.1 q.2-q.1.2)‖ ≤ r})) |
| apply measurable_measure_prodMk_left |
| exact measurableSet_le (by unfold labels; fun_prop) measurable_const |
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| lemma lintegral_ellipsoidMass {d k : ℕ} (ν : Measure (Vec d)) [IsProbabilityMeasure ν] |
| {A : Rows d k} (hA : LinearIndependent ℝ A) (r : ℝ) : |
| (∫⁻ y, ellipsoidMass ν A y r ∂volume) = |
| ENNReal.ofReal (gramVolume A) * volume {y : Labels k | labelNorm y ≤ r} := by |
| let S : Set (Labels k × Vec d) := {z | ‖liftLabel A (labels A z.2-z.1)‖ ≤ r} |
| have hS : MeasurableSet S := measurableSet_le (by unfold labels; fun_prop) measurable_const |
| have hs := (Measure.prod_apply (μ := volume) (ν := ν) hS).symm.trans |
| (Measure.prod_apply_symm (μ := volume) (ν := ν) hS) |
| change (∫⁻ y, ν ((Prod.mk y) ⁻¹' S) ∂volume) = _ |
| rw [hs] |
| calc |
| _ = ∫⁻ _x : Vec d, ENNReal.ofReal (gramVolume A) * |
| volume {y : Labels k | labelNorm y ≤ r} ∂ν := by |
| apply lintegral_congr |
| intro x |
| change volume {y : Labels k | ‖liftLabel A (labels A x-y)‖ ≤ r} = _ |
| simp_rw [norm_liftLabel_sub_comm A (labels A x)] |
| exact label_ellipsoid_translate_volume hA (labels A x) r |
| _ = _ := by simp |
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| lemma normalized_density_integral_le {Y : Type*} [MeasurableSpace Y] (μ : Measure Y) |
| {a h : Y → ℝ≥0∞} (ha : Measurable a) (hh : Measurable h) |
| (hat : ∀ y, a y ≠ ∞) : |
| (∫⁻ y, (a y)⁻¹*h y ∂μ.withDensity a) ≤ ∫⁻ y, h y ∂μ := by |
| rw [lintegral_withDensity_eq_lintegral_mul _ ha (show Measurable (fun y => (a y)⁻¹*h y) from ha.inv.mul hh)] |
| apply lintegral_mono |
| intro y |
| simp only [Pi.mul_apply] |
| rw [← mul_assoc] |
| by_cases haz : a y = 0 |
| · simp [haz] |
| · rw [ENNReal.mul_inv_cancel haz (hat y), one_mul] |
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| lemma normalized_inverse_density_integral {Y : Type*} [MeasurableSpace Y] (μ : Measure Y) |
| {a : Y → ℝ≥0∞} (ha : Measurable a) (hat : ∀ y, a y ≠ ∞) : |
| (∫⁻ y, (a y)⁻¹ ∂μ.withDensity a) = μ {y | a y ≠ 0} := by |
| rw [lintegral_withDensity_eq_lintegral_mul _ ha (show Measurable (fun y => (a y)⁻¹) from ha.inv)] |
| rw [← lintegral_indicator_one (show MeasurableSet {y | a y ≠ 0} from |
| (measurableSet_eq_fun ha measurable_const).compl)] |
| apply lintegral_congr |
| intro y |
| by_cases haz : a y = 0 |
| · simp [haz] |
| · simp [haz,ENNReal.mul_inv_cancel haz (hat y)] |
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| lemma normalized_ellipsoidMass_integral_le {d k : ℕ} {f : Vec d → ℝ≥0∞} |
| (hf : Measurable f) {L : ℝ≥0∞} (hLt : L ≠ ∞) (hfL : ∀ x, f x ≤ L) |
| [IsProbabilityMeasure ((sphereLaw d).withDensity f)] |
| {A : Rows d k} (hA : LinearIndependent ℝ A) (r : ℝ) : |
| (∫⁻ y, (projectedDensity .spherical f A y)⁻¹ * |
| ellipsoidMass ((sphereLaw d).withDensity f) A y r |
| ∂volume.withDensity (projectedDensity .spherical f A)) ≤ |
| ENNReal.ofReal (gramVolume A) * volume {y : Labels k | labelNorm y ≤ r} := by |
| rw [← lintegral_ellipsoidMass ((sphereLaw d).withDensity f) hA r] |
| apply normalized_density_integral_le _ (by fun_prop) (by fun_prop) |
| exact projectedDensity_ne_top .spherical hLt hfL A |
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| end PosteriorReplicas |
| end |
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| end OAI |
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