Differential geometry of smooth families of probability distributions

A smoothly parametrized family of probability distributions forms a manifold. Its differential-geometrical structures are elucidated by introducing a Riemannian metric and one-parameter families of affine connections (-connections). There exists duality between - and - -connections, so that an -flat manifold is automatically - -flat. In an -flat manifold, a natural quasi-distance, called the -divergence can naturally be introduced from the intrinsic dualistic structure. When =-1, this reduces to the Kullback divergence, and when =0 it is the Hellinger distance (which in this case is related to the Riemannian distance). The geometry of -divergence is connected with the - and - -geodesics due to the - and - -connections. It is important in many statistical problems to approximate a distribution by one belonging to a prescribed family of distributions that is closest to the distribution in the sense of the -divergence. This problem of -approximation is solved with the help of the -geodesic and - -geodesic. The geometrical structures of the function space of distributions are also touched upon.