Bayesian Fitting of Dirichlet Type I and II Distributions

Abstract

In his 1986 book, Aitchison explains that compositional data is regularly mishandled in statistical analyses, a pattern that continues to this day. The Dirichlet Type I distribution is a multivariate distribution commonly used to model a set of proportions that sum to one. Aitchinson goes on to lament the difficulties of Dirichlet modelling and the scarcity of alternatives. While he addresses the second of these issues, we address the first. The Dirichlet Type II distribution is a transformation of the Dirichlet Type I distribution and is a multivariate distribution on the positive real numbers with only one more parameter than the number of dimensions. This property of Dirichlet distributions implies advantages over common alternatives as the number of dimensions increase. While not all data is amenable to Dirichlet modelling, there are many cases where the Dirichlet family is the obvious choice. We describe the Dirichlet distributions and show how to fit them using both frequentist and Bayesian methods (we derive and apply two objective priors). The Beta distribution is discussed as a special case. We report a small simulation study to compare the fitting methods. We derive the conditional distributions and posterior predictive conditional distributions. The flexibility of this distribution family is illustrated via examples, the last of which discusses imputation (using the posterior predictive conditional distributions).