Closure of beta and Dirichlet distributions under discrete mixing

We identify a general approach to the construction of discrete mixtures of beta distributions that are themselves beta distributions, and explore a number of examples. The approach extends to discrete mixtures of transformed beta – so-called ‘beta- G ’ – distributions that are themselves beta- G distributions, and to discrete mixtures of Dirichlet distributions that are Dirichlet distributions. Along the way, we make novel observations concerning the Sibuya distribution and introduce some new ‘Winsorized’ distributions.