| import OAI.Probability.PosteriorReplicas.Gaussian.GaussianFiber |
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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 |
| |
| instance matrixRealMeasurableSpace (m n : Type*) : MeasurableSpace (Matrix m n ℝ) := |
| inferInstanceAs (MeasurableSpace (m → n → ℝ)) |
| |
| instance matrixRealBorelSpace (m n : Type*) [Countable m] [Countable n] : |
| BorelSpace (Matrix m n ℝ) := inferInstanceAs (BorelSpace (m → n → ℝ)) |
| |
| attribute [fun_prop] measurable_labels |
| |
| @[fun_prop] lemma measurable_matrix_inv_real (n : ℕ) : |
| Measurable (fun M : Matrix (Fin n) (Fin n) ℝ => M⁻¹) := by |
| simp_rw [Matrix.inv_def, Ring.inverse_eq_inv] |
| exact continuous_id.matrix_det.measurable.inv.smul continuous_id.matrix_adjugate.measurable |
| |
| @[fun_prop] lemma continuous_rowGram {d k : ℕ} : Continuous (@rowGram d k) := by |
| unfold rowGram |
| fun_prop |
| |
| @[fun_prop] lemma measurable_liftLabel {d k : ℕ} : |
| Measurable (fun z : Rows d k × Labels k => liftLabel z.1 z.2) := by |
| unfold liftLabel Matrix.mulVec dotProduct |
| fun_prop |
| |
| @[fun_prop] lemma continuous_liftLabel {d k : ℕ} (A : Rows d k) : |
| Continuous (liftLabel A) := continuous_pairingLift A |
| |
| @[fun_prop] lemma measurable_kernelComponent {d k : ℕ} : |
| Measurable (fun z : Rows d k × Vec d => kernelComponent z.1 z.2) := by |
| unfold kernelComponent |
| fun_prop |
| |
| lemma liftLabel_eq_pairingLift {d k : ℕ} (A : Rows d k) : liftLabel A = pairingLift A := rfl |
| lemma rowGram_eq_gram {d k : ℕ} (A : Rows d k) : rowGram A = Matrix.gram ℝ A := rfl |
| |
| lemma kernelComponent_eq_starProjection {d k : ℕ} {A : Rows d k} |
| (hA : LinearIndependent ℝ A) (x : Vec d) : |
| kernelComponent A x = (Submodule.span ℝ (Set.range A))ᗮ.starProjection x := by |
| rw [Submodule.starProjection_orthogonal_val, starProjection_eq_pairingLift hA] |
| rfl |
| |
| lemma labels_liftLabel {d k : ℕ} {A : Rows d k} (hA : LinearIndependent ℝ A) (y : Labels k) : |
| labels A (liftLabel A y) = y := pairings_pairingLift hA y |
| |
| lemma labels_kernelComponent {d k : ℕ} {A : Rows d k} (hA : LinearIndependent ℝ A) (x : Vec d) : |
| labels A (kernelComponent A x) = 0 := by |
| unfold kernelComponent |
| ext i |
| change ⟪A i, x - liftLabel A (labels A x)⟫ = 0 |
| rw [inner_sub_right] |
| change labels A x i - labels A (liftLabel A (labels A x)) i = 0 |
| rw [labels_liftLabel hA, sub_self] |
| |
| def basePoint {d k : ℕ} (base : BaseLaw) (A : Rows d k) (y : Labels k) (x : Vec d) : Vec d := |
| match base with |
| | .gaussian => liftLabel A y + kernelComponent A x |
| | .spherical => liftLabel A y + |
| (Real.sqrt (1 - ‖liftLabel A y‖ ^ 2) / ‖kernelComponent A x‖) • kernelComponent A x |
| |
| @[fun_prop] lemma measurable_basePoint {d k : ℕ} (base : BaseLaw) : |
| Measurable (fun z : (Rows d k × Labels k) × Vec d => basePoint base z.1.1 z.1.2 z.2) := by |
| cases base <;> unfold basePoint <;> fun_prop |
| |
| lemma baseFiber_eq_map {d k : ℕ} (base : BaseLaw) (A : Rows d k) (y : Labels k) : |
| baseFiber base A y = (stdGaussian (Vec d)).map (basePoint base A y) := by |
| cases base <;> rfl |
| |
| @[fun_prop] lemma measurable_basePoint_fixed {d k : ℕ} (base : BaseLaw) |
| (A : Rows d k) (y : Labels k) : Measurable (basePoint base A y) := by |
| have h := (measurable_basePoint (d := d) (k := k) base).comp |
| (show Measurable (fun x : Vec d => ((A, y), x)) from measurable_const.prodMk measurable_id) |
| exact h |
| |
| instance {d k : ℕ} (base : BaseLaw) (A : Rows d k) (y : Labels k) : |
| IsProbabilityMeasure (baseFiber base A y) := by |
| rw [baseFiber_eq_map] |
| infer_instance |
| |
| lemma labels_basePoint {d k : ℕ} {A : Rows d k} (hA : LinearIndependent ℝ A) |
| (base : BaseLaw) (y : Labels k) (x : Vec d) : labels A (basePoint base A y x) = y := by |
| have h₁ := labels_liftLabel hA y |
| have h₂ := labels_kernelComponent hA x |
| cases base <;> ext i <;> |
| simp only [basePoint, labels, inner_add_right, inner_smul_right, |
| show ⟪A i, liftLabel A y⟫ = y i from congrFun h₁ i, |
| show ⟪A i, kernelComponent A x⟫ = 0 from congrFun h₂ i, mul_zero, add_zero] |
| |
| lemma measurable_parametric_map {Z X Y : Type*} [MeasurableSpace Z] [MeasurableSpace X] |
| [MeasurableSpace Y] (μ : Measure X) [SFinite μ] {F : Z × X → Y} (hF : Measurable F) : |
| Measurable (fun z => μ.map (fun x => F (z, x))) := by |
| apply Measure.measurable_of_measurable_coe |
| intro s hs |
| have he : (fun z => μ.map (fun x => F (z, x)) s) = |
| (fun z => μ ((Prod.mk z) ⁻¹' (F ⁻¹' s))) := by |
| funext z |
| exact Measure.map_apply (hF.comp (measurable_const.prodMk measurable_id)) hs |
| rw [he] |
| exact measurable_measure_prodMk_left (hF hs) |
| |
| @[fun_prop] lemma measurable_baseFiber {d k : ℕ} (base : BaseLaw) : |
| Measurable (fun z : Rows d k × Labels k => baseFiber base z.1 z.2) := by |
| simp_rw [baseFiber_eq_map] |
| exact measurable_parametric_map _ (measurable_basePoint base) |
| |
| def baseKernel {d k : ℕ} (base : BaseLaw) : Kernel (Rows d k × Labels k) (Vec d) where |
| toFun z := baseFiber base z.1 z.2 |
| measurable' := measurable_baseFiber base |
| |
| instance {d k : ℕ} (base : BaseLaw) : IsMarkovKernel (@baseKernel d k base) where |
| isProbabilityMeasure _ := by dsimp [baseKernel]; infer_instance |
|
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| @[fun_prop] lemma measurable_tiltMass {d k : ℕ} (base : BaseLaw) {f : Vec d → ℝ≥0∞} |
| (hf : Measurable f) : Measurable (fun z : Rows d k × Labels k => tiltMass base f z.1 z.2) := by |
| unfold tiltMass |
| exact hf.lintegral_kernel (κ := baseKernel base) |
|
|
| end PosteriorReplicas |
| end |
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| end OAI |
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