| import Mathlib.Probability.Distributions.Gamma | |
| import Mathlib.MeasureTheory.Measure.WithDensity | |
| import Mathlib.MeasureTheory.Measure.Lebesgue.Basic | |
| import Mathlib.Topology.Order.IntermediateValue | |
| import Mathlib.Tactic.Linarith | |
| namespace OAI | |
| namespace GammaFPP | |
| open MeasureTheory ProbabilityTheory Set Filter | |
| open scoped Topology | |
| variable {shape rate : ℝ} | |
| lemma gamma_nullSingleton : NullSingletonClass (gammaMeasure shape rate) := by | |
| unfold gammaMeasure | |
| infer_instance | |
| lemma gamma_cdf_continuous (hs : 0 < shape) (hr : 0 < rate) : | |
| Continuous (cdf (gammaMeasure shape rate)) := by | |
| let := isProbabilityMeasure_gammaMeasure hs hr | |
| let := gamma_nullSingleton (shape := shape) (rate := rate) | |
| apply continuous_iff_continuousAt.mpr | |
| intro x | |
| apply (monotone_cdf (gammaMeasure shape rate)).continuousAt_iff_leftLim_eq_rightLim.mpr | |
| rw [StieltjesFunction.rightLim_eq] | |
| have hzero : ENNReal.ofReal (cdf (gammaMeasure shape rate) x - | |
| Function.leftLim (cdf (gammaMeasure shape rate)) x) = 0 := by | |
| rw [← StieltjesFunction.measure_singleton, measure_cdf, measure_singleton] | |
| have hle := ENNReal.ofReal_eq_zero.mp hzero | |
| have hother := (monotone_cdf (gammaMeasure shape rate)).leftLim_le (le_refl x) | |
| linarith | |
| lemma gamma_Ioc_pos (hs : 0 < shape) (hr : 0 < rate) | |
| {x y : ℝ} (hx : 0 ≤ x) (hxy : x < y) : | |
| 0 < gammaMeasure shape rate (Ioc x y) := by | |
| have hmeas : Measurable (gammaPDF shape rate) := | |
| (measurable_gammaPDFReal shape rate).ennreal_ofReal | |
| have hsupport : {z | gammaPDF shape rate z ≠ 0} ∩ Ioc x y = Ioc x y := by | |
| ext z | |
| constructor | |
| · exact fun h => h.2 | |
| · intro hz | |
| refine ⟨?_, hz⟩ | |
| exact ne_of_gt (ENNReal.ofReal_pos.mpr | |
| (gammaPDFReal_pos hs hr (lt_of_le_of_lt hx hz.1))) | |
| apply pos_iff_ne_zero.mpr | |
| intro hz | |
| have hv := (withDensity_apply_eq_zero hmeas).mp hz | |
| rw [hsupport, Real.volume_Ioc, ENNReal.ofReal_eq_zero] at hv | |
| linarith | |
| lemma gamma_cdf_strictMonoOn (hs : 0 < shape) (hr : 0 < rate) : | |
| StrictMonoOn (cdf (gammaMeasure shape rate)) (Ici 0) := by | |
| let := isProbabilityMeasure_gammaMeasure hs hr | |
| intro x hx y _ hxy | |
| have hp := gamma_Ioc_pos hs hr hx hxy | |
| rw [← measure_cdf (gammaMeasure shape rate), StieltjesFunction.measure_Ioc, | |
| ENNReal.ofReal_pos] at hp | |
| exact sub_pos.mp hp | |
| lemma gamma_cdf_zero (hs : 0 < shape) (hr : 0 < rate) : | |
| cdf (gammaMeasure shape rate) 0 = 0 := by | |
| let := isProbabilityMeasure_gammaMeasure hs hr | |
| let := gamma_nullSingleton (shape := shape) (rate := rate) | |
| have he : gammaMeasure shape rate (Iic 0) = gammaMeasure shape rate (Iio 0) := | |
| (measure_congr Iio_ae_eq_Iic).symm | |
| have hz : gammaMeasure shape rate (Iio 0) = 0 := by | |
| rw [gammaMeasure, withDensity_apply _ measurableSet_Iio] | |
| exact lintegral_gammaPDF_of_nonpos (le_refl 0) | |
| rw [cdf_eq_real, measureReal_def, he, hz, ENNReal.toReal_zero] | |
| lemma gamma_cdf_lt_one (hs : 0 < shape) (hr : 0 < rate) | |
| {x : ℝ} (hx : 0 ≤ x) : cdf (gammaMeasure shape rate) x < 1 := by | |
| have hlt := gamma_cdf_strictMonoOn hs hr hx | |
| (show 0 ≤ x + 1 by linarith) (show x < x + 1 by linarith) | |
| exact hlt.trans_le (cdf_le_one (gammaMeasure shape rate) (x + 1)) | |
| /-- Every `u ∈ [0, 1)` has a unique nonnegative Gamma quantile. -/ | |
| lemma gamma_cdf_existsUnique (hs : 0 < shape) (hr : 0 < rate) | |
| {u : ℝ} (hu : 0 ≤ u) (hu1 : u < 1) : | |
| ∃! x : ℝ, 0 ≤ x ∧ cdf (gammaMeasure shape rate) x = u := by | |
| have hev : ∀ᶠ b : ℝ in atTop, 0 ≤ b ∧ u < cdf (gammaMeasure shape rate) b := by | |
| filter_upwards [eventually_ge_atTop (0 : ℝ), | |
| (tendsto_cdf_atTop (gammaMeasure shape rate)).eventually (Ioi_mem_nhds hu1)] | |
| with b hb hbu | |
| exact ⟨hb, hbu⟩ | |
| obtain ⟨b, hb, hbu⟩ := hev.exists | |
| have hu0 : cdf (gammaMeasure shape rate) 0 ≤ u := by | |
| rw [gamma_cdf_zero hs hr] | |
| exact hu | |
| obtain ⟨x, hx, hxu⟩ := intermediate_value_Icc hb | |
| (gamma_cdf_continuous hs hr).continuousOn ⟨hu0, hbu.le⟩ | |
| refine ⟨x, ⟨hx.1, hxu⟩, ?_⟩ | |
| intro y hy | |
| exact (gamma_cdf_strictMonoOn hs hr).injOn hy.1 hx.1 (hy.2.trans hxu.symm) | |
| end GammaFPP | |
| end OAI | |