Page - 283 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 407 Ifweuse thenotation introducedin(24),wecanwrite gij=− [( ∂ ∂θi ) pθ ( ∂ ∂θj ) pϑ D(α)ϕ (pθ ‖ pϑ) ] pϑ=pθ =E′′θ [∂ϕ−1(pθ) ∂θi ∂ϕ−1(pθ) ∂θj ] . It remains toshowthecaseα=±1.Comparing(13)and(23),wecanwrite Dϕ(pθ ‖ pϑ)=E′θ[ϕ−1(pθ)−ϕ−1(pϑ)]. (35) Weuse theequivalentexpressions gij= [( ∂2 ∂θi∂θj ) p Dαϕ(p‖ q) ] q=p = [( ∂2 ∂θi∂θj ) q Dαϕ(p‖ q) ] q=p , whichfollowsfromcondition(32), to infer that gij= [( ∂2 ∂θi∂θj ) pϑ Dϕ(pθ ‖ pϑ) ] pθ=pϑ =−E′θ [∂2ϕ−1(pθ) ∂θi∂θj ] . (36) BecauseD(−1)ϕ (p ‖ q)=D(1)ϕ (q ‖ p)=Dϕ(p ‖ q),weconclude that themetricdefinedby (22) coincideswith themetric inducedbyD(−1)ϕ (· ‖ ·)andD(1)ϕ (· ‖ ·). In generalized statistical manifolds, the generalized Rényi divergenceD(α)ϕ (· ‖ ·) induces aconnectionD(α),whoseChristoffel symbolsΓ(α)ijk aregivenby Γ(α)ijk =− [( ∂2 ∂θi∂θj ) p ( ∂ ∂θk ) q D(α)ϕ (p‖ q) ] q=p . BecauseD(α)ϕ (p ‖ q) =D(−α)ϕ (q ‖ p), it follows thatD(α) andD(−α) aremutuallydual for any α∈ [−1,1]. Inotherwords,Γ(α)ijk andΓ (−α) ijk satisfy the relation ∂gjk ∂θi = Γ(α)ijk +Γ (−α) ikj . Adevelopment involvingexpression(35) results in Γ(1)ijk =E ′′ θ [∂2ϕ−1(pθ) ∂θi∂θj ∂ϕ−1(pθ) ∂θk ] −E′θ [∂2ϕ−1(pθ) ∂θi∂θj ] E′′θ [ u0 ∂ϕ−1(pθ) ∂θk ] , (37) and Γ(−1)ijk =E ′′ θ [∂2ϕ−1(pθ) ∂θi∂θj ∂ϕ−1(pθ) ∂θk ] +E′′′θ [∂ϕ−1(pθ) ∂θi ∂ϕ−1(pθ) ∂θj ∂ϕ−1(pθ) ∂θk ] −E′′θ [∂ϕ−1(pθ) ∂θj ∂ϕ−1(pθ) ∂θk ] E′′θ [ u0 ∂ϕ−1(pθ) ∂θi ] −E′′θ [∂ϕ−1(pθ) ∂θi ∂ϕ−1(pθ) ∂θk ] E′′θ [ u0 ∂ϕ−1(pθ) ∂θj ] . (38) Forα∈ (−1,1), theChristoffel symbolsΓ(α)ijk canbewrittenasaconvexcombinationofΓ (−1) ijk and Γ(−1)ijk , asasserted in thenext result. Proposition5. TheChristoffel symbolsΓ(α)ijk inducedby thedivergenceD (α) ϕ (· ‖ ·) satisfy the relation Γ(α)ijk = 1−α 2 Γ(−1)ijk + 1+α 2 Γ(1)ijk , forα∈ [−1,1]. (39) 283