Seite - 332 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 421 representation for the Poisson is given inTable 2. Thedivergence parameter—inwhichwehave convexity—isshownforeachλ, as is theso-calledpotential function,whichgenerates thecomplete informationgeometry for thesemodels. Table2.Power-Divergence in thePoissonmodelwithmeanμ,whereλ∗=1−λ. λ α DivergenceDλ(μ1,μ2) DivergenceParameterξ Potential −1 −1 μ1−μ2−μ2(log(μ1)− log(μ2)) ξ= log(μ) exp(ξ) 0 1 μ2−μ1−μ1(log(μ2)− log(μ1)) ξ=μ ξ log(ξ)−ξ λ =0,−1 α =±1 ( λ∗μ1−λ∗μ2−μ2 (( μ1 μ2 )λ∗−1)) λ∗(1−λ∗) ξ= 1 λ∗μ λ∗ (λ∗ξ)1/λ ∗ 1−λ∗ 3.4. ExtendedMultinomialCase In thispaper,weare focusingon the class of log–linearmodelswhere themultinomial is the underlyingclassofdistributions; that is,weconditiononthesamplesize,N,beingfixedintheproduct Poissonspace. Inparticular,wefocusonextendedmultinomials,which includes theclosureof the multinomials, sowehaveaboundary.Dueto theconditioning(which inducescurvature),only the caseswhereλ=0,−1remainBregmandivergences,butall arestilldivergences in thesenseofbeing Csiszár f-divergences [36,37]. Theclosureofanexponentialfamily(e.g., [11,38–40]),anditsapplicationinthetheoryoflog–linear modelshasbeenexplored in [12,13,41,42].Thekeyhere isunderstanding the limitingbehaviour in the natural—α = 1 in the sense of [8]—parameter space. This can be done by considering the polar dual [43], or, alternatively, thedirections of recession—[12] or [42]. Theboundarypolytope determineskeystatisticalpropertiesof themodel, includingthebehaviourof thesamplingdistribution of (functionsof) theMLEandtheshapeof level setsofdivergence functions. Figures 1 and 2 show level sets of the α = ±1 Power-Divergences in the (+1)-affine and (−1)-affine parameters (Panels (a) and (b), respectively) for the k = 2 extended multinomial model. The boundary polytope in this case is a simple triangle “at infinity”, and the shape of this is strongly reflected in the behaviour of the level sets. In Figure 1,we show—in the simplex{ (π0,π1,π2)|∑2i=0πi=1,πi≥0 } —the level sets of the α =−1divergence,which, in theCsiszár f-divergence form, is K(π0,π) := 2 ∑ i=0 log ( π0i πi ) π0i . ThefiguresshowhowinPanel (a), thedirectionsof recessiondominate theshapeof level sets, andin Panel (b) thedualsof thesedirections (i.e., theverticesof thesimplex)eachhavedifferentmaximal behaviour. The lackofconvexityof the level sets inPanel (a)correspondsto the fact that thenatural parametersarenot theaffinedivergenceparameters for thisdivergence, sowedonotexpectconvex behaviour. InPanel (b),wedogetnon-convex level sets, asexpected. Figure2showsthesamestory,but this timefor thedualdivergence, K∗(π,π0) :=K(π0,π). Now, theaffinedivergenceparametersareshowninPanel (a), thenaturalparameters.Wesee that in the limit theshapeof thedivergence isconvergingto thatof thepolarof theboundarypolytope. Ingeneral, localbehaviour isquadratic,butboundarybehaviour ispolygonal. 332