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 ( \(\alpha \) -connections). There exists duality between \(\alpha \) - and \(-\alpha \) -connections, so that an \(\alpha \) -flat manifold is automatically \(-\alpha \) -flat. In an \(\alpha \) -flat manifold, a natural quasi-distance, called the \(\alpha \) -divergence can naturally be introduced from the intrinsic dualistic structure. When \(\alpha =-1\) , this reduces to the Kullback divergence, and when \(\alpha =0\) it is the Hellinger distance (which in this case is related to the Riemannian distance). The geometry of \(\alpha \) -divergence is connected with the \(\alpha \) - and \(-\alpha \) -geodesics due to the \(\alpha \) - and \(-\alpha \) -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 \(\alpha \) -divergence. This problem of \(\alpha \) -approximation is solved with the help of the \(\alpha \) -geodesic and \(-\alpha \) -geodesic. The geometrical structures of the function space of distributions are also touched upon.
