| import Mathlib |
| import OAI.Probability.SKBarriers.SpinGlass.DisorderPairs |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped NNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| def gaussianPair : Measure (ℝ × ℝ) := (gaussianReal 0 1).prod (gaussianReal 0 1) |
| instance gaussianPair_probability : IsProbabilityMeasure gaussianPair := by |
| unfold gaussianPair; infer_instance |
| |
| theorem gaussianPair_fst : HasLaw (Prod.fst : ℝ × ℝ → ℝ) (gaussianReal 0 1) gaussianPair := |
| measurePreserving_fst.hasLaw |
| |
| theorem gaussianPair_snd : HasLaw (Prod.snd : ℝ × ℝ → ℝ) (gaussianReal 0 1) gaussianPair := |
| measurePreserving_snd.hasLaw |
| |
| theorem gaussianPair_independent : IndepFun (Prod.fst : ℝ × ℝ → ℝ) Prod.snd gaussianPair := by |
| exact (indepFun_iff_hasLaw_prodMk_prod gaussianPair_fst gaussianPair_snd).2 HasLaw.id |
| |
| theorem gaussianPair_gaussian : HasGaussianLaw (fun p : ℝ × ℝ => p) gaussianPair := |
| gaussianPair_independent.hasGaussianLaw gaussianPair_fst.hasGaussianLaw gaussianPair_snd.hasGaussianLaw |
| |
| def gaussianMix (a b : ℝ) (p : ℝ × ℝ) : ℝ := a*p.1+b*p.2 |
| |
| def gaussianMixCLM (a b : ℝ) : (ℝ × ℝ) →L[ℝ] ℝ := |
| a • ContinuousLinearMap.fst ℝ ℝ ℝ+b • ContinuousLinearMap.snd ℝ ℝ ℝ |
| |
| theorem gaussianMix_gaussian (a b : ℝ) : HasGaussianLaw (gaussianMix a b) gaussianPair := |
| gaussianPair_gaussian.map_fun (gaussianMixCLM a b) |
| |
| theorem gaussianMix_mean (a b : ℝ) : (∫ p, gaussianMix a b p ∂gaussianPair)=0 := by |
| unfold gaussianMix |
| rw [integral_add (gaussianPair_fst.hasGaussianLaw.integrable.const_mul a) |
| (gaussianPair_snd.hasGaussianLaw.integrable.const_mul b),integral_const_mul,integral_const_mul] |
| have hf := gaussianPair_fst.integral_comp (continuous_id.aestronglyMeasurable) |
| have hs := gaussianPair_snd.integral_comp (continuous_id.aestronglyMeasurable) |
| simp only [Function.comp_def,id_eq,integral_id_gaussianReal] at hf hs |
| rw [hf,hs]; ring |
| |
| theorem gaussianMix_covariance (a b c d : ℝ) : |
| cov[gaussianMix a b,gaussianMix c d;gaussianPair]=a*c+b*d := by |
| have hcross : cov[(Prod.fst : ℝ × ℝ → ℝ),Prod.snd;gaussianPair]=0 := |
| gaussianPair_independent.covariance_eq_zero gaussianPair_fst.hasGaussianLaw.memLp_two |
| gaussianPair_snd.hasGaussianLaw.memLp_two |
| have hf : cov[(Prod.fst : ℝ × ℝ → ℝ),Prod.fst;gaussianPair]=1 := by |
| rw [covariance_self gaussianPair_fst.aemeasurable] |
| have H := gaussianPair_fst.variance_eq |
| simpa only [variance_id_gaussianReal,NNReal.coe_one] using H |
| have hs : cov[(Prod.snd : ℝ × ℝ → ℝ),Prod.snd;gaussianPair]=1 := by |
| rw [covariance_self gaussianPair_snd.aemeasurable] |
| have H := gaussianPair_snd.variance_eq |
| simpa only [variance_id_gaussianReal,NNReal.coe_one] using H |
| unfold gaussianMix |
| rw [SK.covariance_linear_combination gaussianPair_fst.hasGaussianLaw.memLp_two |
| gaussianPair_snd.hasGaussianLaw.memLp_two gaussianPair_fst.hasGaussianLaw.memLp_two |
| gaussianPair_snd.hasGaussianLaw.memLp_two,hf,hs,hcross,covariance_comm Prod.snd,hcross] |
| ring |
| |
| theorem gaussianMix_law (a b : ℝ) (hab : a^2+b^2=1) : |
| HasLaw (gaussianMix a b) (gaussianReal 0 1) gaussianPair := by |
| have hg := gaussianMix_gaussian a b |
| refine ⟨hg.aemeasurable,?_⟩ |
| rw [hg.map_eq_gaussianReal,gaussianMix_mean,← covariance_self hg.aemeasurable, |
| gaussianMix_covariance] |
| have he : a*a+b*b=1 := by nlinarith [hab] |
| rw [he]; norm_num |
| |
| def gaussianRotate (θ : ℝ) (p : ℝ × ℝ) : ℝ × ℝ := |
| (gaussianMix (Real.cos θ) (Real.sin θ) p, |
| gaussianMix (-Real.sin θ) (Real.cos θ) p) |
| |
| theorem gaussianRotate_law (θ : ℝ) : HasLaw (gaussianRotate θ) gaussianPair gaussianPair := by |
| have hg : HasGaussianLaw (gaussianRotate θ) gaussianPair := |
| gaussianPair_gaussian.map_fun ((gaussianMixCLM (Real.cos θ) (Real.sin θ)).prod |
| (gaussianMixCLM (-Real.sin θ) (Real.cos θ))) |
| have hi := hg.indepFun_of_covariance_eq_zero (by rw [gaussianMix_covariance]; ring) |
| exact hi.hasLaw_prod |
| (gaussianMix_law _ _ (by nlinarith [Real.sin_sq_add_cos_sq θ])) |
| (gaussianMix_law _ _ (by nlinarith [Real.sin_sq_add_cos_sq θ])) |
| |
| theorem gaussianRotate_inverse (θ : ℝ) (p : ℝ × ℝ) : |
| gaussianRotate (-θ) (gaussianRotate θ p)=p := by |
| ext <;> simp only [gaussianRotate,gaussianMix,Real.cos_neg,Real.sin_neg] |
| · calc |
| _=(Real.sin θ^2+Real.cos θ^2)*p.1 := by ring |
| _=p.1 := by rw [Real.sin_sq_add_cos_sq]; ring |
| · calc |
| _=(Real.sin θ^2+Real.cos θ^2)*p.2 := by ring |
| _=p.2 := by rw [Real.sin_sq_add_cos_sq]; ring |
| |
| end SK.Analytic |
| |
| end |
| end |
| |
| end OAI |
| |
