Advances and considerations of Dirichlet with an emphasis on entropy

Abstract

The Dirichlet distribution is a cornerstone probabilistic consideration when working with data on the unit simplex. This thesis studies advances of several Dirichlet generalizations such as the Dirichlet generator, noncentral Dirichlet (including meaningful reparameterizations) and two particular finite mixtures of the Dirichlet distribution. Key statistical characteristics of these Dirichlet generalizations are studied together with data fitting examples to highlight the methodological- and computational advantages that these considerations offer. Subsequently, these Dirichlet generalizations are considered in a Bayes framework as priors conjugate to a multinomial likelihood with K distinct classes. An emphasis on entropy follows to determine closed-form estimators for the Tsallis-, generalized Mathai-, and Abe entropies subject to these Dirichlet considerations as priors. These estimators are particularly convenient following from complete product moments of the derived posterior distributions to access the underlying probabilities of the K-dimensional classes, p. These entropies are incorporated within an explorative estimation approach using real economic data and a prior selection method is illustrated to suggest a suitable prior for the consideration of the practitioner. The contributions of this thesis to fundamental multivariate statistical theory as well as the information-theoretic environment (via considered entropies) are meaningful and substantiated.