Information-geometrical characterization of statistical models which are statistically equivalent to probability simplexes

The probability simplex is the set of all probability distributions on a finite set and is the most fundamental object in the finite probability theory. In this paper we give a characterization of statistical models on finite sets which are statistically equivalent to probability simplexes in terms of $\alpha$-families including exponential families and mixture families. The subject has a close relation to some fundamental aspects of information geometry such as $\alpha$-connections and autoparallelity.