Ivancevic_Applied-Diff-Geom

334 Applied Differential Geometry: A Modern Introductionthen for any function ψ = ψ(h) with ψ(0) = 0, ψ(h) > 0 for h > 0, theprobability divergence on M is defined asD(x, y) := ψ(d(x, y)). (3.162)Recall that a Finsler metric L = L(x, y) is a function of tangent vectorsy at a point x ∈ M, with the following properties:L(x, ty) = t 2 L(x, y) for t > 0, (3.163)g ij (x, y) := 1 ∂ 2 L2 ∂y i (x, y) > 0,∂yj F x (y) := √ L(x, y), F x (u + v) ≤ F x (u) + F x (v).This means that there is an inner product g y at a pint x ∈ M, such thatg y (u, v) = g ij (x, y)u i v j ,so that our Finsler metric L(x, y) ∈ M, given by (3.163), becomesL(x, y) = g y (u, v) = g ij (x)y i y j .Therefore, in a special case when g ij (x, y) = g ij (x) are independent of y, theFinsler metric L(x, y) becomes a standard Riemannian metric g ij (x)y i y j .In this way, all the material from the previous subsection can be generalizedto Finsler geometry.Now, D(x, y) ∈ M, given by (3.162), is called the regular divergence, if2D(x, x + y) = L(x, y) + 1 2 L x k(x, y)yk + 1 3 H(x, y) + o(|y|3 ),where H = H(x, y) ∈ M is homogenous function of degree 3 in y, i.e.,H(x, ty) = t 3 H(x, y) for t > 0.A pair {L, H} ∈ M is called a Finsler information structure [Shen (2005)].In a particular case when L(x, y) = g ij (x)y i y j is a Riemannian metric,and H(x, y) = H ijk (x)y i y j y k is a polynomial, then we have affine informationstructure {L, H} ∈ M, which is described by a family of affine connections,called α−connections by [Amari (1985); Amari and Nagaoka (2000)].However, in general, the induced information structure {L, H} ∈ M is notaffine, i.e., L(x, y) is not Riemannian and H(x, y) is not polynomial.