Page - 370 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 98 3.2. Sampling fromL(Y¯,σ) ThecurrentsectionpresentsageneralmethodforsamplingfromtheLaplacedistributionL(Y¯,σ). Thismethodrelies inpartonthe followingtransformationproperty. Proposition1. LetYbea randomvariable inPm. Forall A∈GL(m), Y∼L(Y¯,σ) =⇒ Y ·A∼L(Y¯ ·A,σ) whereY ·AisgivenbyEquation (9). Proof. Letϕ :Pm→Rbeatest function. IfY∼L(Y¯,σ)andZ=Y ·A, thentheexpectationofϕ(Z) isgivenby: E[ϕ(Z)]= ∫ Pm ϕ(X ·A)p(X|Y¯,σ)dv(X) = ∫ Pm ϕ(X)p(X ·A−1|Y¯,σ)dv(X) where the equality is a result of Equation (15). However, p(X ·A−1|Y¯,σ) = p(X|Y¯ ·A,σ) by Equation(10),whichproves theproposition. Thefollowingalgorithmdescribeshowtosample fromL(Y¯,σ)where0<σ<σm. For this, it is first requiredtosample fromthedensity ponRmdefinedby: p(r)= cm ζm(σ) e− |r| σ∏ i<j sinh (|ri−rj| 2 ) , r=(r1, · · · ,rm). ThiscanbedonebyausualMetropolisalgorithm[30]. It isalsorequiredtosample fromtheuniformdistributiononO(m), thegroupof realorthogonal m×mmatrices. This canbedonebygeneratingA, anm×mmatrix,whoseentries are i.i.d. with normaldistributionN(0,1), then theorthogonalmatrixU, in thedecomposition A=UTwithT upper triangular, isuniformlydistributedonO(m) [29] (p. 70). Sampling fromL(Y¯,σ) cannowbe describedas follows. Algorithm1SamplingfromL(Y¯,σ). 1: Generate i.i.d. samples (r1, · · · ,rm)∈Rmwithdensity p 2: GenerateU fromauniformdistributiononO(m) 3: X←U†diag(er1, · · · ,erm)U 4: Y←X.Y¯12 Note that the lawofX inStep3 isL(I,σ); theproofof this fact isgiven inAppendixD.Finally, the lawofY inStep4 isL(I.Y¯12 = Y¯,σ)bypropositionEquation(1). 3.3. Estimationof Y¯ andσ Thecurrentsectionconsidersmaximumlikelihoodestimationof theparameters Y¯andσ, based onindependentobservationsY1, . . . ,YN fromtheRiemannianLaplacedistributionL(Y¯,σ). Themain resultsarecontainedinPropositions2and3below. Proposition2states theexistenceanduniquenessof themaximumlikelihoodestimates YˆN and σˆN of Y¯andσ. Inparticular, themaximumlikelihoodestimate YˆN of Y¯ is theRiemannianmedianof Y1, . . . ,YN,definedbyEquation(11).Numerical computationof YˆN willbeconsideredandcarriedout usingaRiemanniansub-gradientdescentalgorithm[8]. 370