| import OAI.Analysis.FiniteFactor.Sampling.SecondMoment |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal |
| noncomputable section |
|
|
| /-- At each retained starting coordinate, the source weight is exactly the |
| law of the displacement v times the distance to the endpoint. -/ |
| theorem samplingWeight_displacement (h α z : ℝ) |
| (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) |
| (hz : z ∈ Set.Ico (0 : ℝ) 1) (hcut : α ≤ samplingDistance z) : |
| unitIntervalMeasure.withDensity (fun t => ENNReal.ofReal (samplingWeight false h α z t)) = |
| (samplingVelocityMeasure h).map (fun v => samplingDistance z * v) := by |
| let d := samplingDistance z |
| have hd : 0 < d := hα.trans_le hcut |
| have hL : 0 < h*d := mul_pos hh hd |
| have hd1 : d ≤ 1-z := min_le_right _ _ |
| have hL1 : h*d ≤ 1 := by nlinarith [hz.1] |
| have he : (fun t => ENNReal.ofReal (samplingWeight false h α z t)) = |
| (Set.Ioo 0 (h*d)).indicator (fun _ => ENNReal.ofReal (1/(h*d))) := by |
| funext t |
| simp only [samplingWeight,Bool.false_eq_true,ite_false,hcut,true_and,Set.indicator,Set.mem_Ioo,d] |
| split_ifs <;> simp |
| rw [he,withDensity_indicator measurableSet_Ioo,withDensity_const] |
| unfold unitIntervalMeasure |
| rw [Measure.restrict_restrict measurableSet_Ioo, |
| Set.inter_eq_left.mpr (show Set.Ioo 0 (h*d) ⊆ Set.Ico (0 : ℝ) 1 from |
| fun t ht => ⟨ht.1.le,ht.2.trans_le hL1⟩),restrict_Ioo_eq_restrict_Ioc] |
| unfold samplingVelocityMeasure ProbabilityTheory.cond |
| rw [Measure.map_smul _ (by fun_prop)] |
| have ha := map_interval_affine 0 h d 0 hd |
| simp only [mul_zero,add_zero] at ha |
| rw [ha,smul_smul,Real.volume_Ioc,sub_zero] |
| congr 1 |
| rw [← ENNReal.ofReal_inv_of_pos hh,← ENNReal.ofReal_mul (inv_nonneg.mpr hh.le)] |
| congr 1 |
| dsimp [d] |
| field_simp |
| simp [mul_comm] |
|
|
| /-- Exact substitution t=v*Delta(z), at each retained starting point. -/ |
| theorem samplingWeight_transport_integral (h α z : ℝ) |
| (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) |
| (hz : z ∈ Set.Ico (0 : ℝ) 1) (hcut : α ≤ samplingDistance z) |
| (F : ℝ → ℝ≥0∞) (hF : Measurable F) : |
| (∫⁻ t, ENNReal.ofReal (samplingWeight false h α z t)*F (z+t) ∂unitIntervalMeasure) = |
| ∫⁻ v, F (samplingTransport v z) ∂samplingVelocityMeasure h := by |
| have hw : Measurable (fun t => ENNReal.ofReal (samplingWeight false h α z t)) := |
| ((samplingWeight_measurable false h α).comp (measurable_const.prodMk measurable_id)).ennreal_ofReal |
| have hFz : Measurable (fun t => F (z+t)) := hF.comp (by fun_prop) |
| have hi := lintegral_withDensity_eq_lintegral_mul unitIntervalMeasure hw hFz |
| simp only [Pi.mul_apply] at hi |
| rw [← hi, |
| samplingWeight_displacement h α z hh hh1 hα hz hcut, |
| lintegral_map hFz (by fun_prop)] |
| apply lintegral_congr |
| intro v |
| simp [samplingTransport,mul_comm] |
|
|
| /-- Discarding the cutoff only increases the weighted endpoint law. -/ |
| theorem samplingWeight_transport_le (h α z : ℝ) |
| (hh : 0 < h) (hh1 : h < 1) (hα : 0 < α) |
| (hz : z ∈ Set.Ico (0 : ℝ) 1) (F : ℝ → ℝ≥0∞) (hF : Measurable F) : |
| (∫⁻ t, ENNReal.ofReal (samplingWeight false h α z t)*F (z+t) ∂unitIntervalMeasure) ≤ |
| ∫⁻ v, F (samplingTransport v z) ∂samplingVelocityMeasure h := by |
| by_cases hc : α ≤ samplingDistance z |
| · exact (samplingWeight_transport_integral h α z hh hh1 hα hz hc F hF).le |
| · simp [samplingWeight,hc] |
|
|
| end |
| end FiniteFactor |
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| end OAI |
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