| import OAI.Probability.PosteriorReplicas.Geometry.SphereSubspace |
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| namespace OAI |
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| 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] |
|
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| 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 |
|
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| 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 |
|
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| 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 |
|
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| end OAI |
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