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a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | import OAI.Probability.PosteriorReplicas.Geometry.LabelEllipsoids
namespace OAI
open MeasureTheory ProbabilityTheory Set Matrix InnerProductSpace Filter Metric
open scoped ENNReal NNReal RealInnerProductSpace Topology
noncomputable section
namespace PosteriorReplicas
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]
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]
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}
@[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
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
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]
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)]
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
end PosteriorReplicas
end
end OAI
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