Page - 316 - in Differential Geometrical Theory of Statistics

Entropy2017,19, 7 Sinceadifferentialofapower function isalsoapower function,wecangiveacharacterization forescortdistributions. Proposition1. Suppose that Sq is a q-exponential family defined by (1). Then the n-th escort distribution is givenby the n-th differential of q-exponential function. That is, by settingu(t) = (expq t) ′,wehave the following formula: pq(x;θ) = expq ( n ∑ i=1 θiFi(x)−ψ(θ) ) , Pq(x;θ)=Pq,(1)(x;θ) = u ( n ∑ i=1 θiFi(x)−ψ(θ) ) , P˜q(x;θ)=Pq,(2)(x;θ) = u ′ ( n ∑ i=1 θiFi(x)−ψ(θ) ) , ... ... Pq,(n)(x;θ) = u (n−1) ( n ∑ i=1 θiFi(x)−ψ(θ) ) , ... ... Proof. Sinceaq-exponential function isexpq(x)=(1+(1−q))1/(1−q), itsdifferential isgivenby u(x)= 1−q 1−q(1+(1−q)x) 1 1−q−1=(1+(1−q)x) q 1−q ={expq x}q. Therefore,weobtainPq(x;θ)=u ( ∑ni=1θ iFi(x)−ψ(θ) ) . Byinduction, then-thdifferentialofu(x)coincideswiththen-thescortdistributionPq,(n),whichis givenbyEquation(5). 4. StatisticalManifoldsandTheirGeneralizedConformalStructures In this section,weus reviewthegeometryof statisticalmanifolds. Formoredetails about the geometryofstatisticalmanifolds, see [18,19]. Let (S,g)beaRiemannianmanifoldand∇beatorsion-freeaffineconnectiononS.Wesaythat thetriplet (S,∇,g) isa statisticalmanifold if∇g is totallysymmetric. In thiscase,wecandefineatotally symmetric (0,3)-tensorfieldby C(X,Y,Z) :=(∇Xg)(Y,Z) = Xg(Y,Z)−g(∇XY,Z)−g(Y,∇XZ), where X,Y and Z are arbitrary vector fields on S. The tensor field C is called a cubic form or an Amari–Chentsov tensorfield. Thenotionofstatisticalmanifoldwas introducedbyLauritzen[20].Hecalled the triplet (S,g,C) a statisticalmanifold. In this paper, the definition is followed toKurose [18]. Though these two definitions aredifferent, theother statisticalmanifold structure canbeobtained fromagivenone, However, themotivationfor thenotionofconformalequivalenceusing (S,g,C) isdifferent fromthat oneusing (S,∇,g),whichwewilldiscuss in the latterpartof this section. Foragivenstatisticalmanifold (S,∇,g),wecandefineanother torsion-freeaffineconnection∇∗ onSby Xg(Y,Z)= g(∇XY,Z)+g(Y,∇∗XZ). 316