| 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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