Page - 272 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 407 Forinstance, themetric inducedbyϕ-divergenceandthe(ρ,τ)-generalizationoftheFisher–Raometric, for thechoicesρ=ϕ−1 and f = ρ−1,differbyaconformal factor. Among many attempts to generalize Kullback–Leibler divergence, Rényi divergence [24] is one of the most successful, having found many applications [25]. In the present paper, weproposeageneralizationofRényidivergence,whichweusetodefineafamilyofα-connections. This generalization is based on an interpretation of Rényi divergence as a kind of normalizing function. TogeneralizeRényidivergence,weconsideredfunctionssatisfyingsomesuitableconditions. Toafunction forwhich these conditionshold,wegive thenameof ϕ-function. In a limiting case, the generalizedRényi divergence reduces to the ϕ-divergence. In [17,18], the ϕ-divergence gives rise toapairofdualconnectionsD(−1) andD(1).Weshowthat theconnectionD(α) inducedbythe generalizationofRényidivergencesatisfies theconvexcombinationD(α) = 1−α2 D (−1)+ 1+α2 D (1). Eguchi in [26] investigatedageometrybasedonanormalizingfunctionsimilar to theoneused in thegeneralizationofRényidivergence. In [26], resultswerederivedsupposingthat thisnormalizing functionexists; conditions for itsexistencewerenotgiven. In thepresentpaper, theexistenceof the normalizingfunction isensuredbyconditions involvedin thedefinitionofϕ-functions. Therestof thepaper isorganizedas follows. InSection2,ϕ-functionsare introducedandsome propertiesarediscussed. TheRényidivergence isgeneralized inSection3.Weinvestigate inSection4 thegeometry inducedbythegeneralizationofRényidivergence. Section4.2providesevidenceof the roleof thegeneralizedRényidivergence inϕ-families. 2. ϕ-Functions Rényidivergenceisdefinedintermsoftheexponentialfunction(tobemoreprecise, thelogarithm). AwayofgeneralizingRényidivergence is to replace theexponential functionbyanother function, whichsatisfiessomesuitableconditions. Toafunctionforwhichtheseconditionshold,wegive the nameϕ-function. In this section,wedefineandinvestigatesomepropertiesofϕ-functions. Let (T,Σ,μ)beameasurespace.Althoughwedonot restrictouranalysis toaparticularmeasure space, the reader can thinkofT as the set of realnumbersR,Σas theBorelσ-algebraonR, andμ as the Lebesgue measure. We can also consider T to be a discrete set, a case in which μ is the countingmeasure. Wesaythatϕ:R→ (0,∞) isaϕ-function if the followingconditionsaresatisfied: (a1) ϕ(·) is convex; (a2) limu→−∞ϕ(u)=0andlimu→∞ϕ(u)=∞; (a3) thereexistsameasurable functionu0: T→ (0,∞) suchthat∫ T ϕ(c+λu0)dμ<∞, forallλ>0, (1) foreachmeasurable function c: T→Rsatisfying∫T ϕ(c)dμ=1. Thankstocondition(a3),wecangeneralizeRényidivergenceusingϕ-functions. Theseconditions appeared first at [12] where the authors constructed non-parametric ϕ-families of probability distributions.Weremarkthat ifT isfinite, condition(a3) isalwayssatisfied. Examplesof functionsϕ:R→ (0,∞) satisfying(a1)–(a3)abound.Anexampleofgreat relevance is the exponential function ϕ(u) = exp(u), which satisfies conditions (a1)–(a3) with u0 = 1T. Anotherexampleofϕ-function is theKaniadakis’κ-exponential [12,27,28]. Example1. TheKaniadakis’κ-exponentialexpκ :R→ (0,∞) forκ∈ [−1,1] isdefinedas expκ(u)= { (κu+ √ 1+κ2u2)1/κ, ifκ =0, exp(u), ifκ=0, 272