Page - 318 - in Differential Geometrical Theory of Statistics

Entropy2017,19, 7 Proposition3. LetSq={p(x;θ)}beaq-exponential family. Thenthenormalization functionψ(θ) is convex. Proof. Setu(x)=(expqx) ′ and∂i= ∂/∂θi. Thenwehave ∂ip(x;θ) = u ( ∑θkFk(x)−ψ(θ) ) (Fi(x)−∂iψ(θ)), ∂i∂jp(x;θ) = u′ ( ∑θkFk(x)−ψ(θ) ) (Fi(x)−∂iψ(θ))(Fj(x)−∂jψ(θ)) − u ( ∑θkFk(x)−ψ(θ) ) ∂i∂jψ(θ). (9) Since∂i ∫ Ωp(x;θ)dx= ∫ Ω∂ip(x;θ)dx=0and ∫ Ω∂i∂jp(x;θ)dx=0,wehave Zq(p) = ∫ Ω {(p(x;θ)}qdx= ∫ Ω u ( ∑θkFk(x)−ψ(θ) ) dx, ∂i∂jψ(θ) = 1 Zq(p) ∫ Ω u′ ( ∑θkFk(x)−ψ(θ) ) (Fi(x)−∂iψ(θ))(Fj(x)−∂jψ(θ))dx. (10) Foranarbitraryvector c= t(c1,c2, . . . ,cn)∈Rn, sinceZq(p)>0andu′′(x)>0,wehave n ∑ i,j=1 cicj(∂i∂jψ(θ)) = 1 Zq(p) ∫ Ω u′′ ( n ∑ k=1 θkFk(x)−ψ(θ) ){ n ∑ i=1 ci(Fi(x)−∂iψ(θ)) }2 dx ≥ 0. This implies that theHessianmatrix (∂i∂jψ(θ)) is semi-positivedefinite. Weassumethatψ is strictlyconvex in thispaper.Under thisassumption,wecaninducemany geometric structures foraq-exponential family. Sinceψ is strictlyconvex,wecandefineaRiemannianmetricandacubic formby gqij(θ) := ∂i∂jψ(θ), Cqijk(θ) := ∂i∂j∂kψ(θ). Wecallgq andCq aq-Fishermetricandaq-cubic form, respectively [23,24]. Sincegq isaHessianof a functionψ,gq isaHessianmetric, andψ is thepotentialofgq withrespect to thenatural coordinate {θi} [25]. Forafixedrealnumberα, set gq ( ∇q(α)X Y,Z ) := gq ( ∇q(0)X Y,Z ) − α 2 Cq(X,Y,Z) , (11) where∇q(0) is theLevi-Civitaconnectionwithrespect togq. Sincegq isaHessianmetric, fromstandard arguments inHessiangeometry [25],∇q(e) :=∇q(1) and∇q(m) :=∇q(−1) areflataffineconnections andmutuallydualwithrespect togq. Therefore, the triplets (Sq,∇q(e),gq)and (Sq,∇q(m),gq)areflat statisticalmanifolds,andthequadruplet (Sq,gq,∇q(e),∇q(m)) isaduallyflatspace. Underq-expectations,wehavethe followingproposition(cf. [10]). Proposition4. ForSq aq-exponential family, (1)Setηi=Eescq,p[Fi(x)]. Then{ηi} is a∇q(m)-affinecoordinate systemsuch that gq ( ∂ ∂θi , ∂ ∂ηj ) = δ j i. (2)Setφ(η)=Eescq,p[logq p(x;θ)], thenφ(η) is thepotential of g q with respect to{ηi}. 318