Page - 321 - in Differential Geometrical Theory of Statistics

Entropy2017,19, 7 Bydifferentiatingtheaboveequation,wecandefinemutuallydual torsion-freeaffineconnections ∇M(e) and∇M(m): ΓM(e)ij,k (θ) := ∫ Ω (∂i∂j lnq pθ)(∂kpθ)dx, ΓM(m)ij,k (θ) := ∫ Ω (∂k lnq pθ)(∂i∂jpθ)dx, whereΓM(e)ij,k andΓ M(m) ij,k are theChristoffel symbolsof∇M(e) and∇M(m)of thefirstkind, respectively. It is known that gM is aHessianmetric, and thequadruplet (Sq,gM,∇M(e),∇M(m)) is aduallyflat space. Inaddition,anaturalparameter{θi} isa∇M(e)-affinecoordinatesysem.Therefore, thecubic formfor (Sq,∇M(e),gM) is CMijk(θ)=Γ M(m) ij,k (θ). (19) We remark that the statistical manifold structure (Sq,∇M(e),gM) is induced from a β-divergence [17,26] (oradensitypowerdivergence [27]): D1−q(p,r) := ∫ Ω { p(x) p(x)1−q−r(x)1−q 1−q − p(x)2−q−r(x)2−q 2−q } dx. (20) Theorem3. Forthestatisticalmanifoldstructure(Sq,∇M(e),gM), theescortrepresentationsoftheRiemannian metric gM andthe cubic formCM aregivenas follows: gMij (θ) = ∫ Ω (∂i lnq pθ)(∂j lnq pθ)Pq(x;θ)dx, (21) CMijk(θ) = ∫ Ω (∂i lnq pθ)(∂j lnq pθ)(∂k lnq pθ)P˜q(x;θ)dx. (22) Proof. For the Riemannian metric gM, since ∂ipθ = (∂i lnq pθ)Pq(x;θ), we immediately obtain Equation(21) fromthedefinitionofgM. Letusconsider theexpressionforcubic form(22). Theq-score function∂i lnq pθ isunbiasedunder theq-expectation. In fact, Eq,p[∂i lnq pθ]= ∫ Ω (∂j lnq pθ)Pq(x;θ)dx= ∫ Ω ∂jpθdx=0. FromEquation(19),weobtain CMijk(θ) = Γ M(m) ij,k (x;θ) = ∫ Ω (∂k lnq pθ)(∂i∂jpθ)dx = ∫ Ω (∂k lnq pθ)∂i {( ∂j lnq pθ ) Pq(x;θ) } dx = −∂ijψ(θ) ∫ Ω (∂k lnq pθ)Pq(x;θ)dx+ ∫ Ω (∂k lnq pθ)(∂j lnq pθ){∂iPq(x;θ)}dx = ∫ Ω (∂k lnq pθ)(∂j lnq pθ)(∂i lnq pθ)P˜q(x;θ)dx. WeremarkthatNaudts [5]gaveanothergeneralizationofFishermetricgN,which isdefinedby gNij (θ) := ∫ Ω 1 Pescq (x;θ) (∂ipθ)(∂jpθ)dx, 321