| import OAI.Probability.PosteriorReplicas.Kernels.KernelPi |
| import OAI.Probability.PosteriorReplicas.Kernels.ReplicaMeasure |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal |
| noncomputable section |
| namespace PosteriorReplicas |
|
|
| @[fun_prop] lemma measurable_tiltedFiber {d k : ℕ} (base : BaseLaw) |
| {f : Vec d → ℝ≥0∞} (hf : Measurable f) : |
| Measurable (fun z : Data d k => tiltedFiber base f z.1 z.2) := by |
| apply Measure.measurable_of_measurable_coe |
| intro s hs |
| have ht := measurable_tiltMass (d := d) (k := k) base hf |
| have hm : Measurable (fun z : Data d k => (baseFiber base z.1 z.2).withDensity f s) := by |
| simp only [withDensity_apply _ hs] |
| simpa only [lintegral_indicator hs, baseKernel, Kernel.coe_mk] using |
| (hf.indicator hs).lintegral_kernel (κ := baseKernel base) |
| have he : (fun z : Data d k => tiltedFiber base f z.1 z.2 s) = |
| fun z => if tiltMass base f z.1 z.2 = 0 then (Measure.dirac (0 : Vec d)) s |
| else (tiltMass base f z.1 z.2)⁻¹ * (baseFiber base z.1 z.2).withDensity f s := by |
| funext z |
| unfold tiltedFiber |
| split_ifs <;> simp [Measure.smul_apply] |
| rw [he] |
| exact Measurable.ite (measurableSet_eq_fun ht measurable_const) measurable_const (ht.inv.mul hm) |
|
|
| def posteriorKernel {d k : ℕ} (base : BaseLaw) {f : Vec d → ℝ≥0∞} (hf : Measurable f) : |
| Kernel (Data d k) (Vec d) where |
| toFun z := tiltedFiber base f z.1 z.2 |
| measurable' := measurable_tiltedFiber base hf |
|
|
| lemma posteriorKernel_markov {d k : ℕ} (base : BaseLaw) {f : Vec d → ℝ≥0∞} (hf : Measurable f) |
| {L : ℝ≥0∞} (hLt : L ≠ ∞) (hfL : ∀ x, f x ≤ L) : |
| IsMarkovKernel (@posteriorKernel d k base f hf) := |
| ⟨fun z => tiltedFiber_probability base hLt hfL z.1 z.2⟩ |
|
|
| def dataLaw {d k : ℕ} (ν : Measure (Vec d)) : Measure (Data d k) := |
| ((gaussianRows d k).prod ν).map (fun z => (z.1,labels z.1 z.2)) |
|
|
| instance {d k : ℕ} (ν : Measure (Vec d)) [IsProbabilityMeasure ν] : |
| IsProbabilityMeasure (dataLaw (k := k) ν) := by |
| unfold dataLaw |
| infer_instance |
|
|
| lemma posterior_disintegration_lintegral {d k : ℕ} {A : Rows d k} |
| (hA : LinearIndependent ℝ A) (hk : k < d) (base : BaseLaw) |
| {f : Vec d → ℝ≥0∞} (hf : Measurable f) {L : ℝ≥0∞} (hLt : L ≠ ∞) |
| (hfL : ∀ x, f x ≤ L) {H : Labels k × Vec d → ℝ≥0∞} (hH : Measurable H) : |
| (∫⁻ x, H (labels A x,x) ∂(base.measure d).withDensity f) = |
| ∫⁻ y, projectedDensity base f A y * |
| ∫⁻ x, H (y,x) ∂tiltedFiber base f A y := by |
| rw [weighted_base_lintegral hA hk base hf (show Measurable (fun x => H (labels A x,x)) by |
| unfold labels; fun_prop)] |
| apply lintegral_congr |
| intro y |
| by_cases hb : baseDensity base A y = 0 |
| · simp [projectedDensity,hb] |
| have he : (∫⁻ x, f x * H (labels A x,x) ∂baseFiber base A y) = |
| ∫⁻ x, H (y,x) ∂(baseFiber base A y).withDensity f := by |
| rw [lintegral_withDensity_eq_lintegral_mul _ hf (show Measurable (fun x => H (y,x)) by fun_prop)] |
| simp only [Pi.mul_apply] |
| rw [baseFiber_eq_map, |
| lintegral_map (show Measurable (fun x => f x * H (labels A x,x)) by unfold labels; fun_prop) |
| (measurable_basePoint_fixed base A y), |
| lintegral_map (show Measurable (fun x => f x * H (y,x)) by fun_prop) |
| (measurable_basePoint_fixed base A y)] |
| simp only [labels_basePoint hA] |
| rw [he, ← tiltMass_smul_tiltedFiber base hf hLt hfL A y, lintegral_smul_measure] |
| simp only [projectedDensity, smul_eq_mul, mul_assoc] |
|
|
| lemma dataLaw_density {d k : ℕ} (hd : 3 ≤ d) (hk : k < d) (base : BaseLaw) |
| {f : Vec d → ℝ≥0∞} (hf : Measurable f) {L : ℝ≥0∞} (hLt : L ≠ ∞) |
| (hfL : ∀ x, f x ≤ L) : |
| dataLaw (k := k) ((base.measure d).withDensity f) = |
| ((gaussianRows d k).prod volume).withDensity (fun z => projectedDensity base f z.1 z.2) := by |
| let : NeZero d := ⟨by omega⟩ |
| have := bounded_withDensity_finite (base.measure d) hLt hfL |
| apply Measure.ext_of_lintegral |
| intro H hH |
| rw [dataLaw, lintegral_map hH (by unfold labels; fun_prop), lintegral_prod _ (by unfold labels; fun_prop), |
| lintegral_withDensity_eq_lintegral_mul _ (measurable_projectedDensity base hf) hH, |
| lintegral_prod _ (by fun_prop)] |
| apply lintegral_congr_ae |
| filter_upwards [ae_gaussian_rows_full_rank hd hk] with A hA |
| have h := posterior_disintegration_lintegral hA hk base hf hLt hfL |
| (H := fun z : Labels k × Vec d => H (A,z.1)) (by fun_prop) |
| have (y : Labels k) := tiltedFiber_probability base hLt hfL A y |
| simpa using h |
|
|
| end PosteriorReplicas |
| end |
|
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| end OAI |
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|
