Seite - 257 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 277 Usingthe fact that thefirst termin Dˆϕ(pφ,pφT)doesnotdependonφ, so itdoesnotcount in the arginf definingφk+1,weeasilyget (7). Thesameapplies for thecaseof (3). Fornotational simplicity, fromnowon,weredefineDψ withanormalizationbyn, i.e., Dψ(φ,φk)= 1 n n ∑ i=1 ∫ X ψ ( hi(x|φ) hi(x|φk) ) hi(x|φk)dx. (10) Hence,oursetofalgorithmsis redefinedby: φk+1=arginf φ Dˆϕ(pφ,pφT)+Dψ(φ,φ k). (11) Wewill see later that this iteration forces thedivergence todecrease and that, under suitable conditions, it converges toa (local)minimumof Dˆϕ(pφ,pφT). It results thatalgorithm(11)beingaway tocalculateboth theMDϕDE(4)andthekernel-basedMDϕDE(5). 3. SomeConvergencePropertiesofφk Weshowherehow, according to somepossible situations, onemayproveconvergenceof the algorithmdefinedby(11). Letφ0 beagiven initialization,anddefine Φ0 :={φ∈Φ : Dˆϕ(pφ,pφT)≤ Dˆϕ(pφ0,pφT)}, whichwesuppose tobeasubsetof int(Φ). The ideaofdefining this set in this context is inherited fromthepaperWu[16],whichprovidedthefirst correctproof ofconvergence for theEMalgorithm. Beforegoinganyfurther,werecall the followingdefinitionofa (generalized)stationarypoint. Definition 1. Let f : Rd → R be a real valued function. If f is differentiable at a point φ∗ such that ∇f(φ∗)=0,we thensay thatφ∗ is a stationarypoint of f. If f isnotdifferentiable atφ∗ but the subgradientof fatφ∗, say∂f(φ∗), exists such that0∈∂f(φ∗), thenφ∗ is calledageneralized stationarypointof f. Remark 1. In the whole paper, the subgradient is defined for any function not necessarily convex (seeDefinition8.3) in [13] formoredetails. Wewillbeusingthe followingassumptions: A0. Functionsφ → Dˆϕ(pφ|pφT),Dψ are lowersemicontinuous; A1. Functionsφ → Dˆϕ(pφ|pφT),Dψ and∇1Dψ aredefinedandcontinuouson, respectively,Φ,Φ×Φ andΦ×Φ; AC. Functionφ →∇Dˆϕ(pφ|pφT) isdefinedandcontinuousonΦ; A2. Φ0 isacompactsubsetof int(Φ); A3. Dψ(φ,φ¯)>0forall φ¯ =φ∈Φ. Recall also thatwe suppose that hi(x|φ)> 0,dx−a.e.We relax the convexity assumption of functionψ.Weonlysuppose thatψ isnonnegativeandψ(t)=0 iff t=1. Inaddition,ψ′(t)=0 if t=1. Continuityanddifferentiabilityassumptionsof functionφ → Dˆϕ(pφ|pφT) for thecaseof (3)canbe easilycheckedusingLebesgue theorems. Thecontinuityassumptionfor thecaseof (2) canbechecked usingTheorem1.17orCorollary10.14 in [13].DifferentiabilitycanalsobecheckedusingCorollary 10.14orTheorem10.31 in thesamebook. InwhatconcernsDψ, continuityanddifferentiabilitycanbe obtainedmerelybyfulfillingLebesguetheoremsconditions.Whenworkingwithmixturemodels,we onlyneedthecontinuityanddifferentiabilityofψandfunctionshi. The later iseasilydeducedfrom regularityassumptionsonthemodel. ForassumptionA2, there isnouniversalmethod, seeSection4.2 foranExample.AssumptionA3canbecheckedusingLemma2in[2]. 257