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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 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 | import OAI.Probability.PosteriorReplicas.Geometry.SphereSubspace
namespace OAI
open MeasureTheory ProbabilityTheory Set Matrix InnerProductSpace Metric
open scoped ENNReal NNReal RealInnerProductSpace Topology
noncomputable section
namespace PosteriorReplicas
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
lemma volume_pairings_on_span {n : ℕ} {v : Fin n → E} (hv : LinearIndependent ℝ v) :
let U := Submodule.span ℝ (Set.range v)
(volume : Measure U).map (fun x : U => fun i => ⟪v i, (x : E)⟫) =
ENNReal.ofReal ((Real.sqrt (Matrix.gram ℝ v).det)⁻¹) • volume := by
let U := Submodule.span ℝ (Set.range v)
let F : U → Fin n → ℝ := fun x i => ⟪v i, (x : E)⟫
let h : (Fin n → ℝ) → ℝ≥0∞ := fun y => gaussianRadialDensity n ‖pairingLift v y‖
have hF : Measurable F := by unfold F; fun_prop
have hh : Measurable h := by unfold h; fun_prop
have hj : 0 < Real.sqrt (Matrix.gram ℝ v).det :=
Real.sqrt_pos.mpr (Matrix.posDef_gram_of_linearIndependent hv).det_pos
have hU : Module.finrank ℝ U = n := by
simpa only [Fintype.card_fin] using finrank_span_eq_card hv
have hl (x : U) : pairingLift v (F x) = (x : E) := by
rw [← starProjection_eq_pairingLift hv]
exact U.starProjection_eq_self_iff.mpr x.2
have hg : (stdGaussian U).map F = (stdGaussian E).map (fun x => fun i => ⟪v i, x⟫) := by
rw [← gaussian_orthogonalProjectionOnto U, Measure.map_map hF (by fun_prop)]
congr 1
funext x i
exact pairing_starProjection U (Submodule.subset_span (Set.mem_range_self i)) x
have hd : (stdGaussian U) = (volume : Measure U).withDensity (h ∘ F) := by
rw [stdGaussian_density, hU]
congr 1
funext x
simp only [Function.comp_apply, h, hl, Submodule.norm_coe]
have he : ((volume : Measure U).map F).withDensity h =
(ENNReal.ofReal ((Real.sqrt (Matrix.gram ℝ v).det)⁻¹) •
(volume : Measure (Fin n → ℝ))).withDensity h := by
rw [← map_withDensity_comp _ hF hh, ← hd, hg, gaussian_pairings_density hv,
withDensity_smul_measure, ← withDensity_smul _ hh]
congr 1
funext y
change ENNReal.ofReal _ = ENNReal.ofReal _ * gaussianRadialDensity n ‖pairingLift v y‖
rw [gaussianRadialDensity, ← ENNReal.ofReal_mul (by positivity)]
congr 1
ring
have hz (y : Fin n → ℝ) : h y ≠ 0 := by
dsimp [h, gaussianRadialDensity]
positivity
have ht (y : Fin n → ℝ) : h y ≠ ∞ := ENNReal.ofReal_ne_top
have hinv := congrArg (fun μ : Measure (Fin n → ℝ) => μ.withDensity (fun y => (h y)⁻¹)) he
rw [withDensity_inv_same hh (Filter.Eventually.of_forall hz) (Filter.Eventually.of_forall ht),
withDensity_inv_same hh (Filter.Eventually.of_forall hz) (Filter.Eventually.of_forall ht)] at hinv
exact hinv
lemma lintegral_pairingLift {n : ℕ} {v : Fin n → E} (hv : LinearIndependent ℝ v)
{g : E → ℝ≥0∞} (hg : Measurable g) :
let U := Submodule.span ℝ (Set.range v)
∫⁻ x : U, g (x : E) ∂volume =
ENNReal.ofReal ((Real.sqrt (Matrix.gram ℝ v).det)⁻¹) * ∫⁻ y, g (pairingLift v y) := by
let U := Submodule.span ℝ (Set.range v)
have he := congrArg (fun μ : Measure (Fin n → ℝ) => ∫⁻ y, g (pairingLift v y) ∂μ)
(volume_pairings_on_span hv)
rw [lintegral_map (by fun_prop) (by fun_prop), lintegral_smul_measure] at he
change _ = ENNReal.ofReal ((Real.sqrt (Matrix.gram ℝ v).det)⁻¹) *
∫⁻ y, g (pairingLift v y) at he
convert he using 1
apply lintegral_congr
intro x
rw [← starProjection_eq_pairingLift hv, U.starProjection_eq_self_iff.mpr x.2]
def sphericalPairingsDensity {n : ℕ} (v : Fin n → E) (y : Fin n → ℝ) : ℝ≥0∞ :=
if ‖pairingLift v y‖ < 1 then ENNReal.ofReal
(Real.Gamma (((Module.finrank ℝ E : ℕ) : ℝ) / 2) /
(Real.pi ^ ((n : ℝ) / 2) * Real.Gamma ((((Module.finrank ℝ E : ℕ) : ℝ) - n) / 2)) *
(1 - ‖pairingLift v y‖^2) ^ ((((Module.finrank ℝ E : ℕ) : ℝ) - n - 2) / 2) /
Real.sqrt (Matrix.gram ℝ v).det) else 0
omit [FiniteDimensional ℝ E] in
@[fun_prop] lemma measurable_sphericalPairingsDensity [FiniteDimensional ℝ E]
{n : ℕ} (v : Fin n → E) :
Measurable (sphericalPairingsDensity v) := by
unfold sphericalPairingsDensity
exact Measurable.ite (measurableSet_lt (by fun_prop) measurable_const) (by fun_prop) measurable_const
lemma sphere_fiber_integral {n : ℕ} {v : Fin n → E} (hv : LinearIndependent ℝ v)
(hn : n < Module.finrank ℝ E) {g : E → ℝ≥0∞} (hg : Measurable g) :
∫⁻ x, g x ∂uniformDirection E = ∫⁻ y, sphericalPairingsDensity v y *
∫⁻ z, g (pairingLift v y + Real.sqrt (1 - ‖pairingLift v y‖^2) •
normalDirection ((Submodule.span ℝ (Set.range v))ᗮ.starProjection z)) ∂stdGaussian E := by
let U := Submodule.span ℝ (Set.range v)
have hU : Module.finrank ℝ U = n := by
simpa only [Fintype.card_fin] using finrank_span_eq_card hv
have hV : Module.finrank ℝ Uᗮ = Module.finrank ℝ E - n := by
have ha := U.finrank_add_finrank_orthogonal
omega
have hVr : ((Module.finrank ℝ Uᗮ : ℕ) : ℝ) = ((Module.finrank ℝ E : ℕ) : ℝ) - n := by
rw [hV, Nat.cast_sub hn.le]
let : Nontrivial E := Module.nontrivial_of_finrank_pos (by omega : 0 < Module.finrank ℝ E)
let : Nontrivial Uᗮ := Module.nontrivial_of_finrank_pos (by omega : 0 < Module.finrank ℝ Uᗮ)
let C : ℝ := Real.Gamma (((Module.finrank ℝ E : ℕ) : ℝ) / 2) /
(Real.pi ^ ((n : ℝ) / 2) * Real.Gamma ((((Module.finrank ℝ E : ℕ) : ℝ) - n) / 2))
let β : ℝ := (((Module.finrank ℝ E : ℕ) : ℝ) - n - 2) / 2
let G : E → ℝ≥0∞ := fun x => ENNReal.ofReal ((1 - ‖x‖^2)^β) *
∫⁻ z, g (x + Real.sqrt (1 - ‖x‖^2) • normalDirection (Uᗮ.starProjection z)) ∂stdGaussian E
have hG : Measurable G := by unfold G; fun_prop
have hJ : 0 < Real.sqrt (Matrix.gram ℝ v).det :=
Real.sqrt_pos.mpr (Matrix.posDef_gram_of_linearIndependent hv).det_pos
have hC : 0 < C := by
dsimp [C]
have hd : 0 < ((Module.finrank ℝ E : ℕ) : ℝ) := Nat.cast_pos.mpr Module.finrank_pos
have hq : 0 < ((Module.finrank ℝ E : ℕ) : ℝ) - n := by
rw [← hVr]
exact Nat.cast_pos.mpr Module.finrank_pos
positivity
rw [sphere_subspace_gaussian_fiber_integral U hg, sphere_area_ratio, hU, hVr]
change ENNReal.ofReal C * (∫⁻ x : U in ball 0 1, G (x : E)) = _
have hi : (∫⁻ x : U in ball 0 1, G (x : E)) =
ENNReal.ofReal ((Real.sqrt (Matrix.gram ℝ v).det)⁻¹) *
∫⁻ y, (ball (0 : E) 1).indicator G (pairingLift v y) := by
have he := lintegral_pairingLift hv (g := (ball (0 : E) 1).indicator G)
(hG.indicator measurableSet_ball)
convert he using 1
rw [← lintegral_indicator measurableSet_ball]
apply lintegral_congr
intro x
rfl
have hI : Measurable (fun y => (ball (0 : E) 1).indicator G (pairingLift v y)) :=
(hG.indicator measurableSet_ball).comp (continuous_pairingLift v).measurable
rw [hi, ← mul_assoc, ← lintegral_const_mul _ hI]
apply lintegral_congr
intro y
by_cases hy : ‖pairingLift v y‖ < 1
· rw [Set.indicator_of_mem (mem_ball_zero_iff.mpr hy)]
dsimp [G, sphericalPairingsDensity]
rw [ite_eq_left hy, ← mul_assoc, ← ENNReal.ofReal_mul hC.le, ← ENNReal.ofReal_mul (by positivity)]
congr 1
congr 1
dsimp [C, β]
ring
· rw [Set.indicator_of_notMem (mt mem_ball_zero_iff.mp hy)]
simp [sphericalPairingsDensity, hy]
end PosteriorReplicas
end
end OAI
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