Page - 313 - in Differential Geometrical Theory of Statistics

Entropy2017,19, 7 In this paper,we introduce a sequence of escort distributions, thenwe consider a sequential structure of escort expectations. It is known that a deformed exponential family naturally has at least threekindofdifferentstatisticalmanifoldstructures [6,11]. Thenweshowthatsuchstatistical manifoldstructurescanbeobtainedfromasequential structureofescortexpectations. Inparticular, weshowthataFishermetriconaq-exponential familycanbeobtainedfromthedeformedexpectations withrespect to thesecondescortdistribution,andacubic form(oranAmari–Chentsovtensorfield, equivalently) isobtainedfromthedeformedexpectationswithrespect to the thirdescortdistribution. This paper is written based on the proceeding paper [7]. However, this paper focuses on deformedexpectationsofaq-exponential family,whereas thepreviouspaper focusedondeformed independences.Weremarkthatseveralauthorshavebeenstudyingdeformedexpectationsrecently. See [12,13], forexample. 2.DeformedExponentialFamilies In thispaper,weassumethatallobjectsaresmooth forsimplicity. Letusreviewpreliminary facts aboutdeformedexponential functionsanddeformedexponential families. Formoredetails, see [2,6], forexample.Historically,Tsallis [14] introducedthenotionofq-exponential functionandNaudts [5] introducedthenotionofq-exponential family togetherwitha furthergeneralization. Suchahistorical note isprovidedin[2]. LetR++ be the set of all positive real numbers,R++ := {x ∈ R|x> 0}. Let χ be a strictly increasing function from R++ to R++. We define a χ-logarithm function or a deformed logarithm functionby lnχ s := ∫ s 1 1 χ(t) dt. The inverseof lnχ s is calledaχ-exponential functionoradeformed exponential function,which is definedby expχ t :=1+ ∫ t 0 u(s)ds, where the functionu(s) isgivenbyu(lnχ s)=χ(s). Fromnowon,we suppose that χ is a power function, that is, χ(t) = tq. Then thedeformed logarithmandthedeformedexponentialaredefinedby lnq s := s1−q−1 1−q , (s>0), expq t :=(1+(1−q)t) 1 1−q , (1+(1−q)t>0). We say that lnq s is a q-logarithm function and expq t is a q-exponential function. In this case, the functionu(s) isgivenby u(s)=(1+(1−q)s) q 1−q ={expq s}q. Bytakingalimitq→1, thesefunctionscoincidewiththestandardlogarithmlnsandthestandard exponentialexpt, respectively. AstatisticalmodelSq is calledaq-exponential family if Sq := { p(x,θ) ∣∣∣∣∣p(x;θ)= expq [ n ∑ i=1 θiFi(x)−ψ(θ) ] , θ∈Θ⊂Rn } , (1) whereF1(x), . . . ,Fn(x)are functionsonasamplespaceΩ, θ= t(θ1, . . . ,θn) isaparameter, andψ(θ) is thenormalizationwithrespect to theparameter θ. Undersuitableconditions,Sq is regardedasa manifoldwithalocalcoordinatesystem{θ1, . . . ,θn}. Inthiscase,wecall{θi}anaturalcoordinatesystem. 313