Page - 320 - in Differential Geometrical Theory of Statistics

Entropy2017,19, 7 Theorem 2 (cf. [10,24]). For a q-exponential family Sq, two statistical manifolds (Sq,gF,∇(2q−1)) and (Sq,g,∇q(e))are1-conformally equivalent. Inparticular, an invariant statisticalmanifold (Sq,gF,∇(2q−1)) is 1-conformallyflat. Riemannianmetrics andcubic formshave the followingrelations: gqij(θ) = q Zq(p) gFij(θ), (16) Cqijk(θ) = q Zq(p) (2q−1)CFijk(θ) − q Zq(p) { gFij∂k lnZq(p)+g F jk(θ)∂i lnZq(p)+g F ki(θ)∂j lnZq(p) } . (17) Proof. The results were essentially obtained in [10]. However, we give a simpler proof for Equations (16)and(17).Thekeyideaisasequenceofescortdistributionsandtheescortrepresentations ofgF andCF inTheorem1. FromEquation(10),wedirectlyobtain theconformalequivalencerelation(16)usingtheescort representationofgF in (12). Bydifferentiating(9)andtakinganintegration,weobtain 0 = ∫ Ω u′′ ( ∑θlFl(x)−ψ(θ) ) (Fi(x)−∂iψ(θ))(Fj(x)−∂jψ(θ))(Fk(x)−∂kψ(θ))dx − ∫ Ω u′ ( ∑θlFl(x)−ψ(θ) ) (Fk(x)−∂kψ(θ))∂i∂jψ(θ)dx − ∫ Ω u′ ( ∑θlFl(x)−ψ(θ) ) (Fi(x)−∂iψ(θ))∂j∂kψ(θ)dx − ∫ Ω u′ ( ∑θlFl(x)−ψ(θ) ) (Fj(x)−∂jψ(θ))∂k∂iψ(θ)dx − ∫ Ω u ( ∑θlFl(x)−ψ(θ) ) ∂i∂j∂kψ(θ)dx. SinceZq(p)= ∫ ΩPq(x;θ)dx,wehave ∂iZq(p)= ∂i ∫ Ω Pq(x;θ)dx= ∫ Ω ∂iPq(x;θ)dx= ∫ Ω P˜q(x;θ)(Fi(x)−∂iψ(θ))dx. Fromtheescort representationofCF in (13), andProposition1,weobtainEquation (17) since gqij(θ)= ∂i∂jψ(θ)andC q ijk(θ)= ∂i∂j∂kψ(θ). Weremarkthat thecubic formof (Sq,gF,∇(2q−1)) isnotCF but (2q−1)CF. The difference of a α-divergence and a q-relative entropy is only the normalization q/Zq(p). This implies thatanormalizationforprobabilitydensity imposesageneralizedconformalchangefora statisticalmodel. In thenextpartof this section, letusconsideranotherstatisticalmanifoldonSq (cf. [6,17,26]). Recall thataFishermetricgF has the followingrepresentation: gFij(θ)= ∫ Ω (∂i lnpθ)(∂jpθ)dx. In information geometry, ∂i lnpθ is called an e-representation (exponential representation) of pθ, and∂jpθ is calledam-representation (mixture representation). Intuitively,∂i lnpθ and∂jpθ areregarded as tangentvectorsonastatisticalmodel.HenceaFishermetric is regardedasaL2-innerproductof e- andm-representations. Letusgeneralize e-andm-representations foraq-exponential family. For pθ ∈Sq,wecall∂i lnq pθ aq-score function. ThenwedefineaRiemannianmetricgM by gMij (θ)= ∫ Ω (∂i lnq pθ)(∂jpθ)dx. (18) 320