Seite - 282 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 407 InProposition2, theexpressionontheright-handsideofEquation(29)definesadivergenceon its own,whichwas investigatedby JunZhang in [19]. Proposition3asserts that the ϕ-divergence Dϕ(pθ ‖ pϑ)coincideswiththeBregmandivergence[31,32]associatedwiththenormalizingfunctionψ forpointsϑandθ inΘ. Becauseψ is convexandattainsaminimumatθ=0, it followsthat ∂ψ ∂θi (θ)=0 atθ=0.Asaresult, equality (30) reduces toDϕ(p‖ pθ)=ψ(θ). 4.3.Geometry InducedbyD(α)ϕ (· ‖ ·) In thissection,weassumethatϕ(·) is continuouslydifferentiableandstrictlyconvex. The latter assumptionguarantees that D(α)ϕ (p‖ q)=0 ifandonly if p= q. (32) ThegeneralizedRényidivergence inducesametricg=(gij) ingeneralizedstatisticalmanifolds P. Thismetric isgivenby gij=− [( ∂ ∂θi ) p ( ∂ ∂θj ) q Dαϕ(p‖ q) ] q=p . (33) Toshowthat thisexpressiondefinesametric,wehave toverify thatgij is invariantunderchange ofcoordinates,and (gij) ispositivedefinite. Thefirstclaimfollowsfromthechainrule. Thepositive definitenessof (gij) isaconsequenceofProposition4,which isgivenbelow. Proposition4. Themetric inducedbyD(α)ϕ (· ‖ ·) coincideswith themetricgivenby (22)or (25). Proof. Fixanyα∈ (−1,1). Applyingtheoperator ( ∂ ∂θj )pϑ to∫ T ϕ(cα)dμ=1, where cα= 1−α2 ϕ −1(pθ)+ 1+α2 ϕ −1(pϑ)+κ(α)u0,weobtain ∫ T (1+α 2 ∂ϕ−1(pϑ) ∂θj + ( ∂ ∂θj ) pϑ κ(α)u0 ) ϕ′(cα)dμ=0, whichresults in ( ∂ ∂θj ) pϑ κ(α)=−1+α 2 ∫ T ∂ϕ−1(pϑ) ∂θj ϕ′(cα)dμ∫ Tu0ϕ ′(cα)dμ . By thestandarddifferentiationrules,wecanwrite ( ∂ ∂θi ) pθ ( ∂ ∂θj ) pϑ κ(α)=−1+α 2 ∫ T[ 1−α 2 ∂ϕ−1(pθ) ∂θi +( ∂ ∂θi )pθκ(α)u0] ∂ϕ−1(pϑ) ∂θj ϕ′′(cα)dμ∫ Tu0ϕ ′(cα)dμ + 1+α 2 ∫ T ∂ϕ−1(pϑ) ∂θj ϕ′(cα)dμ∫ Tu0ϕ ′(cα)dμ ∫ Tu0[ 1−α 2 ∂ϕ−1(pθ) ∂θi +( ∂ ∂θi )pθκ(α)u0]ϕ ′′(cα)dμ∫ Tu0ϕ ′(cα)dμ . (34) Noticing that ∫ T ∂ϕ−1(pϑ) ∂θj ϕ′(cα)dμ = 0 for pϑ = pθ, the second termon the right-hand sideof Equation(34)vanishes,andthen [( ∂ ∂θi ) pθ ( ∂ ∂θj ) pϑ κ(α) ] pϑ=pθ =−1−α 2 4 ∫ T ∂ϕ−1(pθ) ∂θi ∂ϕ−1(pθ) ∂θj ϕ′′(ϕ−1(pθ))dμ∫ Tu0ϕ ′(ϕ−1(pθ))dμ . 282