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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 | import OAI.Probability.PosteriorReplicas.Gaussian.RadialIntegrals
import OAI.Probability.PosteriorReplicas.Geometry.BallJacobian
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 gaussian_orthogonalProjectionOnto (U : Submodule ℝ E) :
(stdGaussian E).map U.orthogonalProjectionOnto = stdGaussian U := by
apply Measure.ext_of_charFun
ext t
rw [charFun_apply, integral_map (by fun_prop) (by fun_prop)]
have he (x : E) : ⟪U.orthogonalProjectionOnto x, t⟫ = ⟪x, (t : E)⟫ := by
change ⟪U.starProjection x, (t : E)⟫ = _
rw [real_inner_comm, pairing_starProjection U t.2, real_inner_comm]
simp_rw [he]
change charFun (stdGaussian E) (t : E) = _
rw [charFun_stdGaussian, charFun_stdGaussian]
rfl
lemma gaussian_subspace_map (U : Submodule ℝ E) :
(stdGaussian U).map (Subtype.val : U → E) = (stdGaussian E).map U.starProjection := by
rw [← gaussian_orthogonalProjectionOnto U, Measure.map_map (by fun_prop) (by fun_prop)]
rfl
lemma gaussian_subspace_sum (U : Submodule ℝ E) :
((stdGaussian U).prod (stdGaussian Uᗮ)).map (fun z => (z.1 : E) + (z.2 : E)) =
stdGaussian E := by
rw [← gaussian_orthogonal_sum U, ← gaussian_subspace_map, ← gaussian_subspace_map,
Measure.map_prod_map _ _ (by fun_prop) (by fun_prop),
Measure.map_map (by fun_prop) (by fun_prop)]
rfl
omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
lemma subspace_sum_norm_sq [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
(U : Submodule ℝ E) (x : U) (z : Uᗮ) :
‖(x : E) + (z : E)‖^2 = ‖x‖^2 + ‖z‖^2 := by
have ho : ⟪(x : E), (z : E)⟫ = 0 :=
(real_inner_comm _ _).trans ((Submodule.mem_orthogonal' U (z : E)).mp z.2 _ x.2)
simp [norm_add_sq_real, ho]
omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
lemma normalDirection_pos_smul [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
{r : ℝ} (hr : 0 < r) (x : E) :
normalDirection (r • x) = normalDirection x := by
simp [normalDirection, norm_smul, Real.norm_eq_abs, abs_of_pos hr,
_root_.mul_inv_rev, smul_smul, hr.ne']
lemma sphere_subspace_gaussian_integral (U : Submodule ℝ E) [Nontrivial E]
{g : E → ℝ≥0∞} (hg : Measurable g) :
∫⁻ x, g x ∂uniformDirection E =
∫⁻ x : U, ∫⁻ z : Uᗮ, g (normalDirection ((x : E) + (z : E)))
∂stdGaussian Uᗮ ∂stdGaussian U := by
rw [← gaussian_normalDirection, lintegral_map hg (by fun_prop),
← gaussian_subspace_sum U, lintegral_map (by fun_prop) (by fun_prop),
lintegral_prod _ (by fun_prop)]
lemma sphere_subspace_polar_integral (U : Submodule ℝ E) [Nontrivial E] [Nontrivial Uᗮ]
{g : E → ℝ≥0∞} (hg : Measurable g) :
∫⁻ y, g y ∂uniformDirection E =
∫⁻ z : sphere (0 : Uᗮ) 1, ∫⁻ r : Ioi (0 : ℝ), ∫⁻ x : U,
gaussianRadialDensity (Module.finrank ℝ U) ‖x‖ *
g (normalDirection ((x : E) + r.1 • ((z : Uᗮ) : E)))
∂volume ∂gaussianPolarRadius Uᗮ ∂(volume : Measure Uᗮ).toSphere := by
rw [sphere_subspace_gaussian_integral U hg]
have hp (x : U) : (∫⁻ z : Uᗮ, g (normalDirection ((x : E) + (z : E))) ∂stdGaussian Uᗮ) =
∫⁻ z : sphere (0 : Uᗮ) 1, ∫⁻ r : Ioi (0 : ℝ),
g (normalDirection ((x : E) + r.1 • ((z : Uᗮ) : E)))
∂gaussianPolarRadius Uᗮ ∂(volume : Measure Uᗮ).toSphere := by
rw [← gaussian_polar_reconstruction (E := Uᗮ), lintegral_map (by fun_prop) (by fun_prop),
lintegral_prod _ (by fun_prop)]
rfl
simp_rw [hp]
rw [lintegral_lintegral_swap (by fun_prop)]
apply lintegral_congr
intro z
rw [lintegral_lintegral_swap (by fun_prop)]
apply lintegral_congr
intro r
rw [stdGaussian_density, lintegral_withDensity_eq_lintegral_mul _ (by fun_prop) (by fun_prop)]
rfl
lemma radial_subspace_integral (U : Submodule ℝ E) [Nontrivial E] [Nontrivial Uᗮ] (s : ℝ) :
∫⁻ r : Ioi (0 : ℝ), ENNReal.ofReal (r.1 ^ Module.finrank ℝ U) *
gaussianRadialDensity (Module.finrank ℝ U) (r.1 * s) ∂gaussianPolarRadius Uᗮ =
ENNReal.ofReal ((1 + s^2) ^ (-((Module.finrank ℝ E : ℕ) : ℝ) / 2)) *
((volume : Measure E).toSphere univ)⁻¹ := by
rw [lintegral_gaussianPolarRadius (E := Uᗮ) (g := fun r : ℝ =>
ENNReal.ofReal (r ^ Module.finrank ℝ U) *
gaussianRadialDensity (Module.finrank ℝ U) (r * s)) (by fun_prop)]
calc
_ = ∫⁻ r in Ioi (0 : ℝ), ENNReal.ofReal (r ^ (Module.finrank ℝ E - 1)) *
ENNReal.ofReal ((2 * Real.pi) ^ (-((Module.finrank ℝ E : ℕ) : ℝ) / 2) *
Real.exp (-(1 + s^2) * r^2 / 2)) := by
apply setLIntegral_congr_fun measurableSet_Ioi
intro r hr
exact radial_power_gaussian_product Module.finrank_pos U.finrank_add_finrank_orthogonal hr.le s
_ = _ := radial_gaussian_integral (by positivity)
lemma scaled_subspace_integral (U : Submodule ℝ E) (z : sphere (0 : Uᗮ) 1)
(r : Ioi (0 : ℝ)) {g : E → ℝ≥0∞} (hg : Measurable g) :
∫⁻ x : U, gaussianRadialDensity (Module.finrank ℝ U) ‖x‖ *
g (normalDirection ((x : E) + r.1 • ((z : Uᗮ) : E))) =
∫⁻ x : U, ENNReal.ofReal (r.1 ^ Module.finrank ℝ U) *
gaussianRadialDensity (Module.finrank ℝ U) (r.1 * ‖x‖) *
g (normalDirection ((x : E) + ((z : Uᗮ) : E))) := by
rw [lintegral_pos_smul r.2 (by fun_prop), ← lintegral_const_mul _ (by fun_prop)]
apply lintegral_congr
intro x
rw [norm_smul, Real.norm_eq_abs, abs_of_pos r.2]
have he : ((r.1 • x : U) : E) + r.1 • ((z : Uᗮ) : E) =
r.1 • ((x : E) + ((z : Uᗮ) : E)) := by simp [smul_add]
rw [he, normalDirection_pos_smul r.2, mul_assoc]
lemma sphere_subspace_slope_integral (U : Submodule ℝ E) [Nontrivial E] [Nontrivial Uᗮ]
{g : E → ℝ≥0∞} (hg : Measurable g) :
∫⁻ y, g y ∂uniformDirection E =
((volume : Measure E).toSphere univ)⁻¹ *
∫⁻ z : sphere (0 : Uᗮ) 1, ∫⁻ x : U,
ENNReal.ofReal ((1 + ‖x‖^2) ^ (-((Module.finrank ℝ E : ℕ) : ℝ) / 2)) *
g (normalDirection ((x : E) + ((z : Uᗮ) : E)))
∂volume ∂(volume : Measure Uᗮ).toSphere := by
rw [sphere_subspace_polar_integral U hg, ← lintegral_const_mul _ (by fun_prop)]
apply lintegral_congr
intro z
simp_rw [scaled_subspace_integral U z _ hg]
rw [lintegral_lintegral_swap (by fun_prop), ← lintegral_const_mul _ (by fun_prop)]
apply lintegral_congr
intro x
rw [lintegral_mul_const _ (by fun_prop), radial_subspace_integral]
ac_rfl
lemma normalDirection_ballStretch (U : Submodule ℝ E) {x : U}
(hx : x ∈ ball (0 : U) 1) (z : sphere (0 : Uᗮ) 1) :
normalDirection (((ballStretch x : U) : E) + ((z : Uᗮ) : E)) =
(x : E) + Real.sqrt (1 - ‖x‖^2) • ((z : Uᗮ) : E) := by
have hb := one_sub_norm_sq_pos hx
have hs := Real.sqrt_pos.mpr hb
have hz : ‖(z : Uᗮ)‖ = 1 := mem_sphere_zero_iff_norm.mp z.2
have hn : ‖(x : E) + Real.sqrt (1 - ‖x‖^2) • ((z : Uᗮ) : E)‖ = 1 := by
have h := subspace_sum_norm_sq U x (Real.sqrt (1 - ‖x‖^2) • (z : Uᗮ))
simp only [Submodule.coe_smul_of_tower, norm_smul, Real.norm_eq_abs,
abs_of_pos hs, hz, mul_one, Real.sq_sqrt hb.le] at h
nlinarith [norm_nonneg ((x : E) + Real.sqrt (1 - ‖x‖^2) • ((z : Uᗮ) : E))]
have he : ((ballStretch x : U) : E) + ((z : Uᗮ) : E) =
(Real.sqrt (1 - ‖x‖^2))⁻¹ •
((x : E) + Real.sqrt (1 - ‖x‖^2) • ((z : Uᗮ) : E)) := by
rw [ballStretch_eq_inv_sqrt hx]
change (Real.sqrt (1 - ‖x‖^2))⁻¹ • (x : E) + ((z : Uᗮ) : E) = _
rw [smul_add, smul_smul, inv_mul_cancel₀ hs.ne', one_smul]
rw [he, normalDirection_smul_sphere (inv_pos.mpr hs) hn]
lemma sphere_subspace_surface_integral (U : Submodule ℝ E) [Nontrivial E] [Nontrivial Uᗮ]
{g : E → ℝ≥0∞} (hg : Measurable g) :
∫⁻ y, g y ∂uniformDirection E =
((volume : Measure E).toSphere univ)⁻¹ *
∫⁻ x : U in ball 0 1,
ENNReal.ofReal ((1 - ‖x‖^2) ^ ((((Module.finrank ℝ Uᗮ : ℕ) : ℝ) - 2) / 2)) *
∫⁻ z : sphere (0 : Uᗮ) 1,
g ((x : E) + Real.sqrt (1 - ‖x‖^2) • ((z : Uᗮ) : E))
∂(volume : Measure Uᗮ).toSphere := by
rw [sphere_subspace_slope_integral U hg]
congr 1
have ht (z : sphere (0 : Uᗮ) 1) :
(∫⁻ x : U, ENNReal.ofReal ((1 + ‖x‖^2) ^ (-((Module.finrank ℝ E : ℕ) : ℝ) / 2)) *
g (normalDirection ((x : E) + ((z : Uᗮ) : E)))) =
∫⁻ x : U in ball 0 1,
ENNReal.ofReal ((1 - ‖x‖^2) ^ ((((Module.finrank ℝ Uᗮ : ℕ) : ℝ) - 2) / 2)) *
g ((x : E) + Real.sqrt (1 - ‖x‖^2) • ((z : Uᗮ) : E)) := by
rw [lintegral_ballStretch]
apply setLIntegral_congr_fun measurableSet_ball
intro x hx
dsimp only
rw [normalDirection_ballStretch U hx z, ← mul_assoc,
← U.finrank_add_finrank_orthogonal, ballStretch_weight hx]
simp_rw [ht]
rw [lintegral_lintegral_swap (by fun_prop)]
apply setLIntegral_congr_fun measurableSet_ball
intro x hx
dsimp only
rw [lintegral_const_mul _ (by fun_prop)]
lemma surface_lintegral_eq_gaussian [Nontrivial E] {g : E → ℝ≥0∞} (hg : Measurable g) :
∫⁻ z : sphere (0 : E) 1, g (z : E) ∂(volume : Measure E).toSphere =
(volume : Measure E).toSphere univ * ∫⁻ x, g (normalDirection x) ∂stdGaussian E := by
have he : (∫⁻ x, g (normalDirection x) ∂stdGaussian E) = ∫⁻ x, g x ∂uniformDirection E := by
rw [← gaussian_normalDirection, lintegral_map hg (by fun_prop)]
rw [he, uniformDirection, lintegral_map hg (by fun_prop), lintegral_smul_measure]
change _ = (volume : Measure E).toSphere univ *
(((volume : Measure E).toSphere univ)⁻¹ *
∫⁻ z : sphere (0 : E) 1, g (z : E) ∂(volume : Measure E).toSphere)
rw [← mul_assoc, ENNReal.mul_inv_cancel
(Measure.measure_univ_ne_zero.mpr (Measure.toSphere_ne_zero volume)) (measure_ne_top _ _), one_mul]
lemma sphere_subspace_gaussian_fiber_integral (U : Submodule ℝ E)
[Nontrivial E] [Nontrivial Uᗮ] {g : E → ℝ≥0∞} (hg : Measurable g) :
∫⁻ y, g y ∂uniformDirection E =
((volume : Measure E).toSphere univ)⁻¹ * (volume : Measure Uᗮ).toSphere univ *
∫⁻ x : U in ball 0 1,
ENNReal.ofReal ((1 - ‖x‖^2) ^ ((((Module.finrank ℝ Uᗮ : ℕ) : ℝ) - 2) / 2)) *
∫⁻ z : E, g ((x : E) + Real.sqrt (1 - ‖x‖^2) •
normalDirection (Uᗮ.starProjection z)) ∂stdGaussian E := by
rw [sphere_subspace_surface_integral U hg, mul_assoc]
congr 1
rw [← lintegral_const_mul _ (by fun_prop)]
apply setLIntegral_congr_fun measurableSet_ball
intro x hx
dsimp only
rw [surface_lintegral_eq_gaussian (E := Uᗮ) (g := fun z =>
g ((x : E) + Real.sqrt (1 - ‖x‖^2) • (z : E))) (by fun_prop)]
have he : (∫⁻ z : Uᗮ, g ((x : E) + Real.sqrt (1 - ‖x‖^2) • ((normalDirection z : Uᗮ) : E))
∂stdGaussian Uᗮ) = ∫⁻ z : E, g ((x : E) + Real.sqrt (1 - ‖x‖^2) •
normalDirection (Uᗮ.starProjection z)) ∂stdGaussian E := by
rw [← gaussian_orthogonalProjectionOnto Uᗮ, lintegral_map (by fun_prop) (by fun_prop)]
rfl
rw [he]
ac_rfl
lemma sphere_area_ratio (U : Submodule ℝ E) [Nontrivial E] [Nontrivial Uᗮ] :
((volume : Measure E).toSphere univ)⁻¹ * (volume : Measure Uᗮ).toSphere univ =
ENNReal.ofReal (Real.Gamma (((Module.finrank ℝ E : ℕ) : ℝ) / 2) /
(Real.pi ^ (((Module.finrank ℝ U : ℕ) : ℝ) / 2) *
Real.Gamma (((Module.finrank ℝ Uᗮ : ℕ) : ℝ) / 2))) := by
have hd : 0 < ((Module.finrank ℝ E : ℕ) : ℝ) := Nat.cast_pos.mpr Module.finrank_pos
have hq : 0 < ((Module.finrank ℝ Uᗮ : ℕ) : ℝ) := Nat.cast_pos.mpr Module.finrank_pos
have he : ((Module.finrank ℝ E : ℕ) : ℝ) =
((Module.finrank ℝ U : ℕ) : ℝ) + ((Module.finrank ℝ Uᗮ : ℕ) : ℝ) := by
exact_mod_cast U.finrank_add_finrank_orthogonal.symm
have hp : Real.pi ^ (((Module.finrank ℝ E : ℕ) : ℝ) / 2) =
Real.pi ^ (((Module.finrank ℝ U : ℕ) : ℝ) / 2) *
Real.pi ^ (((Module.finrank ℝ Uᗮ : ℕ) : ℝ) / 2) := by
rw [he, add_div, Real.rpow_add Real.pi_pos]
rw [sphere_area, sphere_area, ← ENNReal.ofReal_inv_of_pos (by positivity),
← ENNReal.ofReal_mul (by positivity)]
congr 1
rw [hp]
field_simp
end PosteriorReplicas
end
end OAI
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