Page - 284 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 407 Proof. Forα=±1,equality (39) followstrivially. Thus,weassumeα∈ (−1,1). By (34),wecanwrite ( ∂ ∂θi ) pθ ( ∂ ∂θk ) pϑ κ(α)=−1+α 2 ∫ T[ 1−α 2 ∂ϕ−1(pθ) ∂θi +( ∂ ∂θi )pθκ(α)u0] ∂ϕ−1(pϑ) ∂θk ϕ′′(cα)dμ∫ Tu0ϕ ′(cα)dμ + 1+α 2 ∫ T ∂ϕ−1(pϑ) ∂θk ϕ′(cα)dμ∫ Tu0ϕ ′(cα)dμ ∫ Tu0[ 1−α 2 ∂ϕ−1(pθ) ∂θi +( ∂ ∂θi )pθκ(α)u0]ϕ ′′(cα)dμ∫ Tu0ϕ ′(cα)dμ . (40) Applying ( ∂ ∂θj )pθ to the first termon the right-hand side of (40), and then equating pϑ = pθ, weobtain − 1−α 2 4 E′′θ [∂2ϕ−1(pθ) ∂θi∂θj ∂ϕ−1(pθ) ∂θk ] − 1+α 2 ( ∂2 ∂θi∂θj ) pθ κ(α)E′′θ [ u0 ∂ϕ−1(pθ) ∂θk ] − 1−α 2 4 1−α 2 E′′′θ [∂ϕ−1(pθ) ∂θi ∂ϕ−1(pθ) ∂θj ∂ϕ−1(pθ) ∂θk ] + 1−α2 4 1−α 2 E′′θ [∂ϕ−1(pθ) ∂θi ∂ϕ−1(pϑ) ∂θk ] E′′θ [ u0 ∂ϕ−1(pϑ) ∂θj ] . (41) Similarly, ifweapply ( ∂ ∂θj )pθ to thesecondtermontheright-handsideof (40), andmake pϑ= pθ, weget 1−α2 4 1−α 2 E′′θ [∂ϕ−1(pθ) ∂θj ∂ϕ−1(pθ) ∂θk ] E′′θ [ u0 ∂ϕ−1(pθ) ∂θi ] . (42) Collecting(41)and(42),wecanwrite Γ(α)ijk =− 4 1−α2 [( ∂2 ∂θi∂θj ) pθ ( ∂ ∂θk ) pϑ κ(α) ] pθ=pϑ =E′′θ [∂2ϕ−1(pθ) ∂θi∂θj ∂ϕ−1(pθ) ∂θk ] + 1−α 2 E′′′θ [∂ϕ−1(pθ) ∂θi ∂ϕ−1(pθ) ∂θj ∂ϕ−1(pθ) ∂θk ] − 1−α 2 E′′θ [∂ϕ−1(pθ) ∂θj ∂ϕ−1(pθ) ∂θk ] E′′θ [ u0 ∂ϕ−1(pθ) ∂θi ] − 1−α 2 E′′θ [∂ϕ−1(pθ) ∂θi ∂ϕ−1(pϑ) ∂θk ] E′′θ [ u0 ∂ϕ−1(pϑ) ∂θj ] − 1+α 2 E′θ [∂2ϕ−1(pθ) ∂θi∂θj ] E′′θ [ u0 ∂ϕ−1(pθ) ∂θk ] , (43) whereweused ( ∂2 ∂θi∂θj ) pθ κ(α)= 1−α2 4 [( ∂2 ∂θi∂θj ) pθ D(α)ϕ (pθ ‖ pϑ) ] pϑ=pθ = 1−α2 4 gij=−1−α 2 4 E′θ [∂2ϕ−1(pθ) ∂θi∂θi ] . Expression(39) followsfrom(37), (38)and(43). 5.Conclusions In [17,18], the authors introduced a pair of dual connections D(−1) and D(1) induced by ϕ-divergence. The main motivation of the present work was to find a (non-trivial) family of α-divergences,whose inducedα-connectionsareconvexcombinationsofD(−1) andD(1). Asaresult ofourefforts,weproposedageneralizationofRényidivergence. TheconnectionD(α) inducedbythe generalizationofRényidivergencesatisfies the relationD(α) = 1−α2 D (−1)+ 1+α2 D (1). Togeneralize Rényidivergence,wemadeuseofpropertiesofϕ-functions. Thismakesevident the importanceof ϕ-functions in thegeometryofnon-standardmodels. Instandardstatisticalmanifolds,eventhough Amari’sα-divergenceandRényidivergence (withα∈ [−1,1])donotcoincide, they induce thesame familyofα-connections. Thisstrikingresult requires further investigation. Futureworkshouldfocus 284