Page - 279 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 407 (m3) thematrixg=(gij)definedby gij=−E′θ [∂2ϕ−1(pθ) ∂θi∂θj ] , (22) ispositivedefiniteateachθ∈Θ,where E′θ[·]= ∫ T(·)ϕ′(ϕ−1(pθ))dμ∫ Tu0ϕ ′(ϕ−1(pθ))dμ ; (23) (m4) theoperationsof integrationwithrespect toμanddifferentiationwithrespect toθi commute in all calculations foundbelow,whicharerelatedto themetricandconnections. Thematrixg=(gij)equipsPwithametric. Bythechainrule, the tensorrelatedtog=(gij) is invariantunderchangeofcoordinates. The (classical) statisticalmanifold isaparticularcase inwhich ϕ(u)= exp(u)andu0= 1T. Weintroduceanotationsimilar toEquation(23) that involveshigherorderderivativesofϕ(·). Foreachn≥1,wedefine E(n)θ [·]= ∫ T(·)ϕ(n)(ϕ−1(pθ))dμ∫ Tu0ϕ ′(ϕ−1(pθ))dμ . (24) WealsouseE′θ[·],E′′θ [·]andE′′′θ [·] todenoteE(n)θ [·] forn=1,2,3, respectively. Thenotation(24) appears inexpressionsrelatedto themetricandconnections. Using property (m4), we can find an alternate expression for gij aswell as an identification involvingtangentspaces. Thematrixg=(gij)canbeequivalentlydefinedby gij=E′′θ [∂ϕ−1(pθ) ∂θi ∂ϕ−1(pθ) ∂θj ] . (25) As a consequence of this equivalence, the tangent space TpθP can be identified with T˜pθP, the vector space spanned by ∂ϕ −1(pθ) ∂θi , and endowed with the inner product 〈X˜,Y˜〉θ := E′′θ [X˜Y˜]. Themapping ∑ i ai ∂ ∂θi →∑ i ai ∂ϕ−1(pθ) ∂θi definesan isometrybetweenTpθP and T˜pθP. Toverify (25),wedifferentiate ∫ T pθdμ=1,withrespect toθ i, toget 0= ∂ ∂θi ∫ T pθdμ= ∫ T ∂ ∂θi ϕ(ϕ−1(pθ))dμ= ∫ T ∂ϕ−1(pθ) ∂θi ϕ′(ϕ−1(pθ))dμ. (26) Now,differentiatingwithrespect toθj,weobtain 0= ∫ T ∂2ϕ−1(pθ) ∂θi∂θj ϕ′(ϕ−1(pθ))dμ+ ∫ T ∂ϕ−1(pθ) ∂θi ∂ϕ−1(pθ) ∂θj ϕ′′(ϕ−1(pθ))dμ, and then (25) follows. In view of (26), we notice that every vector X˜ belonging to T˜pθP satisfiesE′θ[X˜]=0. The metric g = (gij) gives rise to a Levi–Civita connection ∇ (i.e., a torsion-free, metric connection),whosecorrespondingChristoffel symbolsΓijk aregivenby Γijk := 1 2 (∂gki ∂θj + ∂gkj ∂θi − ∂gij ∂θk ) . (27) 279