| import OAI.Probability.PosteriorReplicas.Geometry.FrameTubes |
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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 |
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| @[fun_prop] lemma continuous_basePoint_label {d k : ℕ} (base : BaseLaw) |
| (A : Rows d k) (x : Vec d) : Continuous (fun y => basePoint base A y x) := by |
| cases base <;> unfold basePoint <;> fun_prop |
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| lemma baseDensity_sphere_clipped {d k : ℕ} (hk : k + 2 < d) {A : Rows d k} |
| (hA : LinearIndependent ℝ A) (y : Labels k) : |
| baseDensity .spherical A y = ENNReal.ofReal |
| (baseProjectionConstant .spherical d k * |
| (max (1 - ‖liftLabel A y‖^2) 0) ^ (((d : ℝ) - k - 2) / 2) / |
| Real.sqrt (rowGram A).det) := by |
| have hβ : 0 < ((d : ℝ) - k - 2) / 2 := by |
| have : (k : ℝ) + 2 < d := by exact_mod_cast hk |
| linarith |
| rw [baseDensity.eq_def, ite_eq_left hA] |
| dsimp only [baseProjectionConstant] |
| by_cases hy : ‖liftLabel A y‖ < 1 |
| · rw [ite_eq_left hy, max_eq_left (by nlinarith [norm_nonneg (liftLabel A y)])] |
| · rw [ite_eq_right hy, max_eq_right (by nlinarith [le_of_not_gt hy]), |
| Real.zero_rpow hβ.ne', mul_zero, zero_div, ENNReal.ofReal_zero] |
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| lemma continuous_baseDensity {d k : ℕ} (hk : k + 2 < d) (base : BaseLaw) |
| {A : Rows d k} (hA : LinearIndependent ℝ A) : Continuous (baseDensity base A) := by |
| cases base |
| · have hβ : 0 ≤ ((d : ℝ) - k - 2) / 2 := by |
| have : (k : ℝ) + 2 < d := by exact_mod_cast hk |
| linarith |
| have he : baseDensity .spherical A = fun y => ENNReal.ofReal |
| (baseProjectionConstant .spherical d k * |
| (max (1 - ‖liftLabel A y‖^2) 0) ^ (((d : ℝ) - k - 2) / 2) / |
| Real.sqrt (rowGram A).det) := funext (baseDensity_sphere_clipped hk hA) |
| rw [he] |
| apply ENNReal.continuous_ofReal.comp |
| apply Continuous.div_const |
| apply Continuous.const_mul |
| exact (Real.continuous_rpow_const hβ).comp (by fun_prop) |
| · have he : baseDensity .gaussian A = fun y => ENNReal.ofReal |
| ((2 * Real.pi) ^ (-(k : ℝ) / 2) * |
| Real.exp (-‖liftLabel A y‖^2 / 2) / Real.sqrt (rowGram A).det) := by |
| funext y |
| rw [baseDensity.eq_def, ite_eq_left hA] |
| rw [he] |
| exact ENNReal.continuous_ofReal.comp (by fun_prop) |
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| lemma base_sample_representation {d k : ℕ} {A : Rows d k} (hA : LinearIndependent ℝ A) |
| (hk : k < d) (base : BaseLaw) : |
| ((volume.withDensity (baseDensity base A)).prod (stdGaussian (Vec d))).map |
| (fun z => basePoint base A z.1 z.2) = base.measure d := by |
| have : NeZero d := ⟨by omega⟩ |
| have hb0 := (measurable_baseDensity (d := d) (k := k) base).comp |
| (show Measurable (fun y : Labels k => (A,y)) from measurable_const.prodMk measurable_id) |
| simp only [Function.comp_def] at hb0 |
| have hb : Measurable (baseDensity base A) := hb0 |
| have hm : Measurable (fun z : Labels k × Vec d => basePoint base A z.1 z.2) := by |
| have h := (measurable_basePoint (d := d) (k := k) base).comp |
| (show Measurable (fun z : Labels k × Vec d => ((A,z.1),z.2)) from |
| (measurable_const.prodMk measurable_fst).prodMk measurable_snd) |
| exact h |
| have : IsProbabilityMeasure (volume.withDensity (baseDensity base A)) := by |
| rw [← projected_base_measure hA hk] |
| infer_instance |
| apply Measure.ext_of_lintegral |
| intro g hg |
| have hgm : Measurable (fun z : Labels k × Vec d => g (basePoint base A z.1 z.2)) := hg.comp hm |
| rw [lintegral_map hg hm, lintegral_prod _ hgm.aemeasurable, |
| lintegral_withDensity_eq_lintegral_mul _ hb (by fun_prop), base_lintegral hA hk base hg] |
| apply lintegral_congr |
| intro y |
| simp only [Pi.mul_apply] |
| rw [baseFiber_eq_map, lintegral_map hg (measurable_basePoint_fixed base A y)] |
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| end PosteriorReplicas |
| end |
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| end OAI |
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