Page - 374 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 98 Section 4.2 considers the problem of order selection for mixtures of Riemannian Laplace distributions. Precisely, thisconsistsoffindingthenumberMofmixturecomponents inEquation(27) that realizes the best representation of a given set of dataY1, . . . ,YN. This problem is solved by computingtheBICcriterion,whichisherefoundinexplicit formforthecaseofmixturesofRiemannian LaplacedistributionsonPm. 4.1. Estimationof theMixtureParameters In thissection,Y1, . . . ,YN are i.i.d. samples fromEquation(27). Basedontheseobservations,an EMalgorithm isproposed to estimate ( μ,Y¯μ,σμ)1≤μ≤M. Thederivationof this algorithmcanbe carriedoutsimilarly to [15]. Toexplainhowthisalgorithmworks,defineforallϑ={( μ,Y¯μ,σμ)}, ωμ(Yn,ϑ)= μ×p(Yn|Y¯μ,σμ) ∑Ms=1 s×p(Yn|Y¯s,σs) , Nμ(ϑ)= N ∑ n=1 ωμ(Yn) (28) Thealgorithmiterativelyupdates ϑˆ={(ˆμ,Yˆμ, σˆμ)} ,whichisanapproximationofthemaximum likelihoodestimateof themixtureparametersϑ=( μ,Y¯μ,σμ)as follows. • Update for ˆμ: Basedonthecurrentvalueof ϑˆ, assignto ˆμ thenewvalue ˆμ= Nμ(ϑˆ) / N. • Update for Yˆμ: Basedonthecurrentvalueof ϑˆ, assignto Yˆμ thevalue: Yˆμ=argminY N ∑ n=1 ωμ(Yn, ϑˆ)d(Y,Yn) (29) • Update for σˆμ: Basedonthecurrentvalueof ϑˆ, assignto σˆμ thenewvalue: σˆμ=Φ(N−1μ (ϑˆ)×∑Nn=1 ωμ(Yn, ϑˆ)d(Yˆμ,Yn)) (30) where the functionΦ isdefinedinProposition1. These threeupdaterulesshouldbeperformedin theaboveorder. Realizationof theupdaterules for ˆμ and σˆμ is straightforward. Theupdaterule for Yˆμ is realizedusingaslightmodificationof the sub-gradientdescentalgorithmdescribed inSection3.2.Moreprecisely, the factor1/Nappearing in Equation(22) isonlyreplacedwithωμ(Yn, ϑˆ)ateach iteration. Inpractice, the initial conditions (ˆμ0,Yˆμ0, σˆμ0) in thisalgorithmwerechosen in the following way. Theweights ( μ0) are uniformand equal to 1/M; (Yˆμ0) are M different observations from the set {Y1,..,YN} chosen randomly; and (σˆμ0) is computed from ( μ0) and (Yˆμ0) according to the rule Equation (30). Since the convergence of the algorithmdepends on the initial conditions, theEMalgorithm is run several times, and thebest result is retained, i.e., theonemaximizing the log-likelihoodfunction. 4.2. TheBayesian InformationCriterion TheBICwas introducedbySchwarz tofindtheappropriatedimensionofamodel thatwillfita givensetofobservations [16]. Since then,BIChasbeenusedinmanyBayesianmodelingproblems wherepriorsarehardtosetprecisely. In largesamplesettings, thefittedmodel favoredbyBICideally correspondsto thecandidatemodel that isaposteriorimostprobable; i.e., themodel that is rendered mostplausiblebythedataathand.Oneof themainfeaturesof theBICis itseasycomputation, since it isonlybasedontheempirical log-likelihoodfunction. 374