| import OAI.Probability.PosteriorReplicas.Measures.EqualLabelVersions |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal Topology |
| noncomputable section |
| namespace PosteriorReplicas |
|
|
| def equalReference {d k m : ℕ} (ν : Measure (Vec d)) : Measure (EqualSpace d k m) := |
| ((Measure.pi (fun _ : Fin (m+1) => ν)).prod (gaussianRows d k)).map rightBasePoint |
| |
| instance {d k m : ℕ} (ν : Measure (Vec d)) [IsProbabilityMeasure ν] : |
| IsProbabilityMeasure (equalReference (k := k) (m := m) ν) := by |
| unfold equalReference |
| infer_instance |
| |
| instance {d k m : ℕ} (ν : Measure (Vec d)) [IsFiniteMeasure ν] : |
| IsFiniteMeasure (equalReference (k := k) (m := m) ν) := by |
| unfold equalReference |
| infer_instance |
| |
| def replicaDensity {d k m : ℕ} (base : BaseLaw) (f : Vec d → ℝ≥0∞) |
| (z : EqualSpace d k m) : ℝ≥0∞ := |
| equalConstant k m * (ENNReal.ofReal (affineVolume z.2.2))⁻¹ ^ k * |
| (projectedDensity base f z.1 z.2.1)⁻¹ ^ m |
| |
| @[fun_prop] lemma measurable_replicaDensity {d k m : ℕ} (base : BaseLaw) |
| {f : Vec d → ℝ≥0∞} (hf : Measurable f) : Measurable (@replicaDensity d k m base f) := by |
| unfold replicaDensity |
| fun_prop |
| |
| def replicaLaw {d k m : ℕ} (base : BaseLaw) (f : Vec d → ℝ≥0∞) : Measure (EqualSpace d k m) := |
| (equalReference ((base.measure d).withDensity f)).withDensity (replicaDensity base f) |
| |
| lemma equalReference_lintegral {d k m : ℕ} (ν : Measure (Vec d)) [IsFiniteMeasure ν] |
| {H : EqualSpace d k m → ℝ≥0∞} (hH : Measurable H) : |
| ∫⁻ z, H z ∂equalReference ν = ∫⁻ u : Tuple d m, ∫⁻ A, |
| H (residualRows u A, labels (residualRows u A) (u 0), u) ∂gaussianRows d k |
| ∂Measure.pi (fun _ => ν) := by |
| rw [equalReference, lintegral_map hH measurable_rightBasePoint, |
| lintegral_prod _ (show AEMeasurable (fun z => H (rightBasePoint z)) _ from |
| (hH.comp measurable_rightBasePoint).aemeasurable)] |
| rfl |
| |
| lemma density_power_cancel (m : ℕ) {a : ℝ≥0∞} (ha : a ≠ ∞) : |
| a^(m+1) * a⁻¹^m = a := by |
| by_cases h0 : a = 0 |
| · simp [h0] |
| · rw [pow_succ, mul_comm (a^m) a, mul_assoc, ← mul_pow, ENNReal.mul_inv_cancel h0 ha, |
| one_pow, mul_one] |
| |
| lemma replicaLaw_lintegral {d k m : ℕ} (hd : k+m+2 ≤ d) (hk : 1 ≤ k) (hm : 1 ≤ m) |
| (base : BaseLaw) {f : Vec d → ℝ≥0∞} (hf : Measurable f) |
| {L : ℝ≥0∞} (hLt : L ≠ ∞) (hfL : ∀ x, f x ≤ L) |
| {H : EqualSpace d k m → ℝ≥0∞} (hH : Measurable H) : |
| ∫⁻ z, H z ∂replicaLaw base f = |
| ∫⁻ A, ∫⁻ y, projectedDensity base f A y * |
| ∫⁻ u, H (A,y,u) ∂Measure.pi (fun _ : Fin (m+1) => tiltedFiber base f A y) |
| ∂volume ∂gaussianRows d k := by |
| let : NeZero d := ⟨by omega⟩ |
| have := bounded_withDensity_finite (base.measure d) hLt hfL |
| have hG : Measurable (fun z : EqualSpace d k m => |
| (projectedDensity base f z.1 z.2.1)⁻¹^m * H z) := by fun_prop |
| have he := equal_label_tilt_identity hd hk hm base hf hLt hfL hG |
| rw [equalLabelRight_residual (by omega) (by omega) base hf hLt hfL hG] at he |
| have hrep : (∫⁻ z, H z ∂replicaLaw base f) = equalLabelLeft base f |
| (fun z : EqualSpace d k m => (projectedDensity base f z.1 z.2.1)⁻¹^m * H z) := by |
| rw [he] |
| rw [replicaLaw, lintegral_withDensity_eq_lintegral_mul _ (measurable_replicaDensity base hf) hH] |
| simp only [Pi.mul_apply] |
| rw [equalReference_lintegral (base.measure d |>.withDensity f) (by fun_prop)] |
| unfold replicaDensity |
| simp only |
| rw [← lintegral_const_mul _ (show Measurable (fun u : Tuple d m => |
| (ENNReal.ofReal (affineVolume u))⁻¹^k * |
| ∫⁻ A, (projectedDensity base f (residualRows u A) |
| (labels (residualRows u A) (u 0)))⁻¹^m * |
| H (residualRows u A, labels (residualRows u A) (u 0), u) ∂gaussianRows d k) by |
| unfold residualRows; fun_prop)] |
| apply lintegral_congr |
| intro u |
| rw [← mul_assoc, ← lintegral_const_mul _ (show Measurable (fun A => |
| (projectedDensity base f (residualRows u A) |
| (labels (residualRows u A) (u 0)))⁻¹^m * |
| H (residualRows u A, labels (residualRows u A) (u 0), u)) by unfold residualRows; fun_prop)] |
| apply lintegral_congr |
| intro A |
| ring |
| rw [hrep] |
| unfold equalLabelLeft |
| apply lintegral_congr |
| intro A |
| apply lintegral_congr |
| intro y |
| dsimp only |
| rw [lintegral_const_mul _ (show Measurable (fun u : Tuple d m => H (A,y,u)) by fun_prop), |
| ← mul_assoc, density_power_cancel m (projectedDensity_ne_top base hLt hfL A y)] |
| |
| lemma lintegral_projectedDensity {d k : ℕ} {A : Rows d k} (hA : LinearIndependent ℝ A) |
| (hk : k < d) (base : BaseLaw) {f : Vec d → ℝ≥0∞} (hf : Measurable f) : |
| (∫⁻ y, projectedDensity base f A y) = (base.measure d).withDensity f univ := by |
| have h := congrArg (fun μ : Measure (Labels k) => μ univ) (projected_measure hA hk base hf) |
| rw [Measure.map_apply (by unfold labels; fun_prop) MeasurableSet.univ, |
| preimage_univ, withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ] at h |
| simpa only [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ] using h.symm |
| |
| lemma replicaLaw_mass {d k m : ℕ} (hd : k+m+2 ≤ d) (hk : 1 ≤ k) (hm : 1 ≤ m) |
| (base : BaseLaw) {f : Vec d → ℝ≥0∞} (hf : Measurable f) |
| {L : ℝ≥0∞} (hLt : L ≠ ∞) (hfL : ∀ x, f x ≤ L) : |
| replicaLaw (k := k) (m := m) base f univ = (base.measure d).withDensity f univ := by |
| rw [← lintegral_one, replicaLaw_lintegral hd hk hm base hf hLt hfL measurable_const] |
| have (A : Rows d k) (y : Labels k) := tiltedFiber_probability base hLt hfL A y |
| simp only [lintegral_one, measure_univ, mul_one] |
| calc |
| _ = ∫⁻ (_ : Rows d k), (base.measure d).withDensity f univ ∂gaussianRows d k := by |
| apply lintegral_congr_ae |
| filter_upwards [ae_gaussian_rows_full_rank (by omega : 3 ≤ d) (by omega : k < d)] with A hA |
| exact lintegral_projectedDensity hA (by omega) base hf |
| _ = _ := by simp |
| |
| lemma replicaLaw_probability {d k m : ℕ} (hd : k+m+2 ≤ d) (hk : 1 ≤ k) (hm : 1 ≤ m) |
| (base : BaseLaw) {f : Vec d → ℝ≥0∞} (hf : Measurable f) |
| {L : ℝ≥0∞} (hLt : L ≠ ∞) (hfL : ∀ x, f x ≤ L) |
| [IsProbabilityMeasure ((base.measure d).withDensity f)] : |
| IsProbabilityMeasure (replicaLaw (k := k) (m := m) base f) := by |
| constructor |
| rw [replicaLaw_mass hd hk hm base hf hLt hfL, measure_univ] |
| |
| end PosteriorReplicas |
| end |
| |
| end OAI |
| |
