This paper presents a new Bayesian approach to texture classification, yielding enhanced performance in the presence of intraclass diversity. From a mathematical point of view, it specifies an original EM algorithm for mixture estimation on Riemannian manifolds, generalising existing, non probabilistic, clustering analysis methods. For texture classification, the chosen feature space is the Riemannian manifold known as the Poincaré half plane, here denoted H, (this is the set of univariate normal distributions, equipped with Rao's distance). Classes are modelled as finite mixtures of Riemannian priors, (Riemannian priors are probability distributions, recently introduced by the authors, which represent clusters of points in H). During the training phase of classification, the EM algorithm, proposed in this paper, computes maximum likelihood estimates of the parameters of these mixtures. The algorithm combines the structure of an EM algorithm for mixture estimation, with a Riemannian gradient descent, for computing weighted Riemannian centres of mass.