| import OAI.Probability.PosteriorReplicas.Geometry.DensityBinsGeometry |
| import OAI.Probability.PosteriorReplicas.Geometry.LabelConvolution |
| import OAI.Probability.PosteriorReplicas.Measures.ReferenceMoments |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Set Matrix InnerProductSpace Filter Metric |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| noncomputable section |
| namespace PosteriorReplicas |
|
|
| def expectedDensity {d k m : ℕ} (f : Vec d → ℝ≥0∞) (u : Tuple d m) : ℝ≥0∞ := |
| ∫⁻ A, projectedDensity .spherical f (residualRows u A) |
| (labels (residualRows u A) (u 0)) ∂gaussianRows d k |
|
|
| @[fun_prop] lemma measurable_expectedDensity {d k m : ℕ} {f : Vec d → ℝ≥0∞} |
| (hf : Measurable f) : Measurable (@expectedDensity d k m f) := by |
| unfold expectedDensity |
| fun_prop |
|
|
| def binEnvelope {d k : ℕ} (ν : Measure (Vec d)) (J : ℕ) (A : Rows d k) (y : Labels k) : ℝ≥0∞ := |
| 2 + ∑ j ∈ Finset.range J, ENNReal.ofReal ((2:ℝ)^(k*(j+1))) * ellipsoidMass ν A y ((1/2:ℝ)^j) |
|
|
| @[fun_prop] lemma measurable_binEnvelope {d k : ℕ} (ν : Measure (Vec d)) [SFinite ν] (J : ℕ) : |
| Measurable (fun z : Data d k => binEnvelope ν J z.1 z.2) := by |
| unfold binEnvelope |
| fun_prop |
|
|
| lemma ae_expectedDensity_le_inverse {d k m : ℕ} (hd : k+m+2 ≤ d) (hk : 1 ≤ k) (hm : 1 ≤ m) |
| {f : Vec d → ℝ≥0∞} (hf : Measurable f) {L : ℝ≥0∞} (hLt : L ≠ ∞) (hfL : ∀ x, f x ≤ L) : |
| ∀ᵐ u : Tuple d m ∂Measure.pi (fun _ => (sphereLaw d).withDensity f), |
| expectedDensity (k := k) f u ≤ ∫⁻ x, ENNReal.ofReal ((tupleDistance u x)⁻¹^k) |
| ∂(sphereLaw d).withDensity f := by |
| let : NeZero d := ⟨by omega⟩ |
| have : IsProbabilityMeasure (sphereLaw d) := by |
| change IsProbabilityMeasure (BaseLaw.spherical.measure d); infer_instance |
| have := bounded_withDensity_finite (sphereLaw d) hLt hfL |
| have h := (equalLabelVersions d k m hk hm (by omega) .spherical f hf ⟨L,hLt,hfL⟩).2.2.2.2 |
| filter_upwards [h] with u hu |
| have hu' : LinearIndependent ℝ (differences u) := hu.1 |
| have he : (fun A : Rows d k => residualRows u A) = |
| fun A i => (differenceSpan u)ᗮ.starProjection (A i) := by |
| funext A i |
| exact kernelComponent_eq_starProjection hu' _ |
| have hv : (fun A : Rows d k => projectedDensity .spherical f A (labels A (u 0))) =ᵐ[constrainedRows k u] |
| (fun A => ballDensityVersion ((sphereLaw d).withDensity f) A (labels A (u 0))) := by |
| filter_upwards [hu.2] with A hA |
| exact hA.2.2.2.2.2.1 |
| rw [constrainedRows_eq_residualRows hu'] at hv |
| have hv' := (ae_map_iff (show Measurable (residualRows u) by fun_prop).aemeasurable |
| (measurableSet_eq_fun (by unfold labels; fun_prop) (by unfold labels; fun_prop))).mp hv |
| unfold expectedDensity |
| rw [lintegral_congr_ae hv'] |
| simp_rw [he] |
| apply expected_ballDensityVersion_le _ _ _ |
| filter_upwards [tupleDistance_pos_ae (by omega) (by omega) u f] with x hx |
| exact norm_ne_zero_iff.mp (ne_of_gt hx) |
|
|
| lemma ae_constrainedRows_mem {d k m : ℕ} (u : Tuple d m) : |
| ∀ᵐ A ∂constrainedRows k u, ∀ i, A i ∈ (differenceSpan u)ᗮ := by |
| unfold constrainedRows |
| have hp : MeasurableSet {A : Rows d k | ∀ i, A i ∈ (differenceSpan u)ᗮ} := by |
| rw [Set.ofPred_forall] |
| exact MeasurableSet.iInter (fun i => (differenceSpan u)ᗮ.closed_of_finiteDimensional.measurableSet.preimage (measurable_pi_apply i)) |
| exact (ae_map_iff (show AEMeasurable (fun A : Rows d k => fun i => |
| (differenceSpan u)ᗮ.starProjection (A i)) _ from |
| (Measurable.of_eval (fun i => (differenceSpan u)ᗮ.starProjection.continuous.measurable.comp (measurable_pi_apply i))).aemeasurable) hp).mpr |
| (ae_of_all _ (fun A i => Submodule.starProjection_apply_mem _ _)) |
|
|
| lemma ae_equalReference_bin_bound {d k m : ℕ} (hd : k+m+2 ≤ d) (hk : 1 ≤ k) (hm : 1 ≤ m) |
| {f : Vec d → ℝ≥0∞} (hf : Measurable f) {L : ℝ≥0∞} (hLt : L ≠ ∞) (hfL : ∀ x, f x ≤ L) |
| [IsProbabilityMeasure ((sphereLaw d).withDensity f)] (J : ℕ) |
| (hcut : (L*ENNReal.ofReal ((72:ℝ)^d))*ENNReal.ofReal ((1/2:ℝ)^J)* |
| (1+(2:ℝ≥0∞)^(k+1)) ≤ 1) : |
| ∀ᵐ z : EqualSpace d k m ∂equalReference ((sphereLaw d).withDensity f), |
| expectedDensity (k := k) f z.2.2 ≤ binEnvelope ((sphereLaw d).withDensity f) J z.1 z.2.1 := by |
| apply ae_equalReference_of_constrained (by omega) (by omega) .spherical hf hLt hfL |
| (measurableSet_le (by fun_prop) (by fun_prop)) |
| filter_upwards [ae_expectedDensity_le_inverse hd hk hm hf hLt hfL, |
| ae_constrained_support hd hk hm .spherical hf hLt hfL] with u hu hs |
| filter_upwards [hs.2, ae_constrainedRows_mem (k := k) u] with A hA hU |
| exact hu.trans (density_inverse_moment_bins hd u hfL J hcut hA.1 hU) |
|
|
| lemma binTerm_volume {k : ℕ} (j : ℕ) : |
| ENNReal.ofReal ((2:ℝ)^(k*(j+1))) * volume {y : Labels k | labelNorm y ≤ (1/2:ℝ)^j} = |
| (2:ℝ≥0∞)^k * volume {y : Labels k | labelNorm y ≤ 1} := by |
| rw [label_closedBall_scale (by positivity), ← mul_assoc, ← ENNReal.ofReal_mul (by positivity)] |
| have he : (2:ℝ)^(k*(j+1)) * ((1/2:ℝ)^j)^k = (2:ℝ)^k := by |
| rw [Nat.mul_comm k (j+1), pow_mul, ← mul_pow] |
| congr 1 |
| rw [pow_succ, div_pow, one_pow, div_eq_mul_inv] |
| field_simp |
| rw [he, ENNReal.ofReal_pow (by norm_num)] |
| norm_num |
|
|
| lemma normalized_binEnvelope_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) (J : ℕ) : |
| (∫⁻ y, (projectedDensity .spherical f A y)⁻¹ * binEnvelope ((sphereLaw d).withDensity f) J A y |
| ∂volume.withDensity (projectedDensity .spherical f A)) ≤ |
| ENNReal.ofReal (gramVolume A) * volume {y : Labels k | labelNorm y ≤ 1} * |
| (2+(J:ℝ≥0∞)*2^k) := by |
| simp only [binEnvelope, mul_add, Finset.mul_sum] |
| rw [lintegral_add_left (by fun_prop), lintegral_mul_const _ (by fun_prop), |
| normalized_inverse_density_integral _ (by fun_prop) (projectedDensity_ne_top .spherical hLt hfL A), |
| lintegral_finsetSum _ (fun j _ => by fun_prop)] |
| calc |
| _ ≤ (ENNReal.ofReal (gramVolume A) * volume {y : Labels k | labelNorm y ≤ 1}) * 2 + |
| ∑ j ∈ Finset.range J, ENNReal.ofReal ((2:ℝ)^(k*(j+1))) * |
| (ENNReal.ofReal (gramVolume A) * volume {y : Labels k | labelNorm y ≤ (1/2:ℝ)^j}) := by |
| apply add_le_add |
| · exact mul_le_mul' (projectedDensity_support_volume hA f) le_rfl |
| · apply Finset.sum_le_sum |
| intro j hj |
| simp only [← Nat.mul_add] |
| simp_rw [mul_left_comm ((projectedDensity .spherical f A _)⁻¹) (ENNReal.ofReal ((2:ℝ)^(k*(j+1))))] |
| rw [lintegral_const_mul _ (by fun_prop)] |
| exact mul_le_mul' le_rfl (normalized_ellipsoidMass_integral_le hf hLt hfL hA _) |
| _ = _ := by |
| simp_rw [mul_left_comm _ (ENNReal.ofReal (gramVolume A)), binTerm_volume] |
| simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul] |
| ring |
|
|
| end PosteriorReplicas |
| end |
|
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| end OAI |
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