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 (α-connections). There exists duality between α- and −α-connections, so that an α-flat manifold is automatically −α-flat. In an α-flat manifold, a natural quasi-distance, called the α-divergence can naturally be introduced from the intrinsic dualistic structure. When α =− 1, this reduces to the Kullback divergence, and whenα = 0 it is the Hellinger distance (which in this case is related to the Riemannian distance). The geometry of α-divergence is connected with Communicated by Nihat Ay. This article is a renewed version of the original manuscript written in 1982 with the same title (METR 82-7, Dept. Math. Eng. and Instr. Phys., Univ. of Tokyo), which is included as a supplementary material: https://static-content.springer.com/esm/art%3A10.1007%2Fs41884-025-00187-y/MediaObjects/41884_ 2025_187_MOESM1_ESM.pdf. While placing the highest possible priority on faithfully reproducing the original (including its incomplete parts), we found it necessary to make changes in the following two respects. The first concerns the format and presentation of the paper. This work is not a reprint of a previously published article, but rather a new publication. Therefore, it must conform to the format expected of a paper published today in the journal Information Geometry. For instance, the authors’ affiliations are given as their current ones, and the acknowledgments at the end refer to the preparation of the present version. The second concerns the content of the paper. In the course of preparing the manuscript for publication, we realized that the original manuscript contained many parts that required correction or annotation. However, making unlimited revisions from a modern perspective would be inconsistent with our aim of publishing the original work. Considering this point, we have made the following two types of corrections and additions to the paper. (a) Edits to the main text: we tried to keep these edits minimal and only included changes that met both of the following two criteria: - The issue is minor, such as typos, awkward wording or simple mathematical errors. - The reason for the change would be clear to anyone comparing it with the original manuscript. (b) Additional explanations in footnotes: when the issue is more substantial, such as mathematical errors that are more than minor, missing conditions, overly concise expressions or potentially misleading exposition, we added explanatory notes in footnotes. (Note: the original manuscript contains no footnote.) We hope that these changes enhance the correctness and the readability of the paper, while preserving its significance as a historical document. Extended author information available on the last page of the article 123 H. Nagaoka, S.-i. Amari the α- and −α-geodesics due to the α- and −α-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 theα-divergence. This problem of α-approximation is solved with the help of the α-geodesic and −α- geodesic. The geometrical structures of the function space of distributions are also touched upon.