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

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