Page - 322 - in Differential Geometrical Theory of Statistics

Entropy2017,19, 7 ThemetricgN is conformallyequivalent togMwithconformal factorZq(pθ)= ∫ Ω{p(x;θ)}qdx. That is, gN(θ)=Zq(pθ)gM(θ). (Seealso [6]). Naudtsgavea furthergeneralizationofFishermetric andheshowedaCramér–Raotypeboundtheorem[5]. 6.ConcludingRemarks In thispaper,we introducedasequenceofescortdistributions. Thenwegaverepresentationsof Riemannianmetricsandcubic formsfromaviewpointof thesequenceofescortdistributions. Inparticular,wecandefinethefollowing(0,2)-tensorfieldsonaq-exponential family. Forpθ ∈Sq, setηi= ∂iψ(θ). (1) Fromthestandardexpectation,weobtain g(0)ij (θ) := Gij(θ) := ∫ Ω (∂i lnq pθ)(∂j lnq pθ)pθdx = Ep[(Fi(x)−ηi)(Fj(x)−ηj)]. The tensorG isacovariancematrix.However,Gmaynotbe important inanomalousstatistics. (2) Fromtheq-expectation,weobtain g(1)ij (θ) := g M ij (θ) = ∫ Ω (∂i lnq pθ)(∂j lnq pθ){pθ}qdx = ∫ Ω (∂i lnq pθ)(∂j lnq pθ)Pq(x;θ)dx = Eq,p[(Fi(x)−ηi)(Fj(x)−ηj)]. TheRiemannianmetricgM isaHessianmetric, andit is inducedfromtheβ-divergence (20). (3) Fromtheexpectationwithrespect to thesecondescortdistribution,weobtain g(2)ij (θ) := g F ij(θ) = ∫ Ω (∂i lnq pθ)(∂j lnq pθ){pθ}2q−1dx = 1 q ∫ Ω (∂i lnq pθ)(∂j lnq pθ)P˜q(x;θ)dx = 1 q Eq,(2),p[(Fi(x)−ηi)(Fj(x)−ηj)]. gqij(θ) = Zq(p) q gFij. TheRiemannianmetric gF is a Fishermetric. Hence gF is invariant to the choiceof reference measureonΩ, but it isnotaHessianmetric. Inaddition, gF is inducedfromtheα-divergence (14). TheconformalRiemannianmetricgq isaq-Fishermetric. It isaHessianmetric, and it is inducedfrom anormalizedTsallis relativeentropy(15). WemaydefineaRiemannianmetricandacubic formfromhigherorderescortexpectations: g(n)ij (θ) := ∫ Ω (∂i lnq pθ)(∂j lnq pθ)Pq,(n)(x;θ)dx, C(n)ij (θ) := ∫ Ω (∂i lnq pθ)(∂j lnq pθ)(∂k lnq pθ)Pq,(n+1)(x;θ)dx. Thenweobtainasequenceofstatisticalmanifoldstructures. (Sq,g(1),C(1)) → (Sq,g(2),C(2)) → ··· → (Sq,g(n),C(n)) → ··· However, the geometric meaning of this sequence is not clear at this moment. Elucidating geometricpropertiesof this sequence isa futureproblem. 322