Page - 317 - in Differential Geometrical Theory of Statistics

Entropy2017,19, 7 Theconnection∇∗ is called thedual connectionof∇withrespect tog. The triplet (S,∇∗,g) isalso astatisticalmanifold,which iscalled thedual statisticalmanifoldof (S,∇,g). Thecubic formisgivenby thedifferenceof twoaffineconnections∇∗ and∇: C(X,Y,Z)= g(∇∗XY−∇XY,Z). Wedefine generalized conformal structures for statisticalmanifolds followed toKurose [18]. Twostatisticalmanifolds (S,∇,g)and (S,∇¯, g¯)are said tobe1-conformally equivalent if thereexists a functionλ :S→R++ suchthat g¯(X,Y) = λg(X,Y), (6) ∇¯XY = ∇XY−g(X,Y)gradg(lnλ), (7) wheregradg(lnλ) is thegradientvectorfieldof lnλwith respect to g, that is, g(X, lnλ) =X(lnλ). We say that (S,∇,g) is 1-conformally flat if (S,∇,g) is locally 1-conformally equivalent to a flat statisticalmanifold. Twostatisticalmanifolds (S,∇,g)and (S,∇¯, g¯)aresaid tobe (−1)-conformally equivalent if there existsa functionλ :S→R++ suchthat g¯(X,Y) = λg(X,Y), ∇¯XY = ∇XY+d(lnλ)(Y)X+d(lnλ)(X)Y, (8) where d(lnλ)(X) = X(lnλ). If two statistical manifolds (S,∇,g) and (S,∇¯, g¯) are 1-conformally equivalent, then their dual statistical manifolds (S,∇∗,g) and (S,∇¯∗, g¯) are (−1)-conformallyequivalent. Proposition2. If two statisticalmanifolds (S,∇,g) and (S,∇¯, g¯) are1-conformally equivalent, then their cubic formshave the followingrelation: 1 λ C¯(X,Y,Z)=C(X,Y,Z)+g(Y,Z)d(lnλ)(X)+g(Z,X)d(lnλ)(Y)+g(X,Y)d(lnλ)(Z). Proof. FromEquations (7)and(8),weobtain ∇¯XY=∇XY+d(lnλ)(Y)X+d(lnλ)(X)Y+g(X,Y)gradg(lnλ). By takinganinnerproductwithrespect tog,weobtain theresult. 5. StatisticalManifoldStructuresonq-ExponentialFamilies In thissection,weconsiderstatisticalmanifoldstructuresonaq-exponential family. It isknown thataq-exponential familynaturallyhasat least threekindsofstatisticalmanifoldstructures (cf. [6,8]). Wereformulate these structures fromtheviewpointof the sequenceof escortdistributions. In this paper,weomit thedetailsabout informationgeometry. See [21,22] for furtherdetails. Firstly, we review basic facts about q-exponential family. Let Sq be a q-exponential family. The normalization ψ(θ) on Sq is convex, but may not be strictly convex. In fact, we obtain the followingproposition. 317