Page - 280 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 407 Usingexpression(25) tocalculate thederivatives in (27),wecanexpress Γijk=E′′θ [∂2ϕ−1(pθ) ∂θi∂θj ∂ϕ−1(pθ) ∂θk ] + 1 2 E′′′θ [∂ϕ−1(pθ) ∂θi ∂ϕ−1(pθ) ∂θj ∂ϕ−1(pθ) ∂θk ] − 1 2 E′′θ [∂ϕ−1(pθ) ∂θi ∂ϕ−1(pθ) ∂θk ] E′′θ [ u0 ∂ϕ−1(pθ) ∂θj ] − 1 2 E′′θ [∂ϕ−1(pθ) ∂θj ∂ϕ−1(pθ) ∂θk ] E′′θ [ u0 ∂ϕ−1(pθ) ∂θi ] + 1 2 E′′θ [∂ϕ−1(pθ) ∂θi ∂ϕ−1(pθ) ∂θj ] E′′θ [ u0 ∂ϕ−1(pθ) ∂θk ] . Aswewill showlater, theLevi–Civitaconnection∇corresponds to theconnectionderivedfrom thedivergenceD(α)ϕ (· ‖ ·)withα=0. 4.2. ϕ-Families Let c: T → R be a measurable function for which p = ϕ(c) is a probability density inPμ. Fixmeasurable functionsu1, . . . ,un : T→R. A (parametric) ϕ-familyFp= {pθ : θ∈Θ}, centeredat p=ϕ(c), isasetofprobabilitydistributions inPμ,whosememberscanbewritten in the form pθ :=ϕ ( c+ n ∑ i=1 θiui−ψ(θ)u0 ) , foreachθ=(θi)∈Θ, (28) whereψ:Θ→ [0,∞) is anormalizing function,which is introduced so that expression (28)defines aprobabilitydistributionbelongingtoPμ. The functionsu1, . . . ,un arenotarbitrary. Theyarechosentosatisfy the followingassumptions: (i) u0,u1, . . . ,un are linearly independent, (ii) ∫ Tuiϕ ′(c)dμ=0,and (iii) thereexists ε>0suchthat ∫ T ϕ(c+λui)dμ<∞, forallλ∈ (−ε,ε). Moreover, thedomainΘ⊆Rn isdefinedas thesetofallvectorsθ=(θi) forwhich ∫ T ϕ ( c+λ n ∑ i=1 θiui ) dμ<∞, for someλ>1. Condition(i) implies that themappingdefinedby(28) isone-to-one.Assumption(ii)makesofψ anon-negative function. Indeed,bytheconvexityofϕ(·), alongwith (ii),wecanwrite ∫ T ϕ(c)dμ= ∫ T [ ϕ(c)+ ( n ∑ i=1 θiui ) ϕ′(c) ] dμ≤ ∫ T ϕ ( c+ n ∑ i=1 θiui ) dμ, which impliesψ(θ)≥0. Bycondition(iii), thedomainΘ isanopenneighborhoodof theorigin. If the setT isfinite, condition(iii) isalwayssatisfied.Onecanshowthat thedomainΘ isopenandconvex. Moreover, the normalizing function ψ is also convex (or strictly convex if ϕ(·) is strictly convex). Conditions (ii) and(iii) alsoappears in thedefinitionofnon-parametricϕ-families. For furtherdetails, werefer to [11,12]. Inaϕ-familyFp, thematrix (gij)givenby(22)or (25)canbeexpressedas theHessianofψ. Ifϕ(·) is strictlyconvex, then(gij) ispositivedefinite. From ∂ϕ−1(pθ) ∂θi =ui− ∂ψ ∂θi , −∂ 2ϕ−1(pθ) ∂θi∂θj =− ∂ 2ψ ∂θi∂θj , it followsthatgij= ∂2ψ ∂θi∂θj . Thenext tworesults showhowthegeneralizationofRényidivergenceandtheϕ-divergenceare relatedto thenormalizingfunction inϕ-families. 280