Improved control of Dirichlet location and scale near the boundary

Dirichlet distributions are commonly used for modeling vectors in a probability simplex. When used as a prior or a proposal distribution, it is natural to set the mean of a Dirichlet to be equal to the location where one wants the distribution to be centered. However, if the mean is near the boundary of the probability simplex, then a Dirichlet distribution becomes highly concentrated either (i) at the mean or (ii) extremely close to the boundary. Consequently, centering at the mean provides poor control over the location and scale near the boundary. In this article, we introduce a method for improved control over the location and scale of Beta and Dirichlet distributions. Specifically, given a target location point and a desired scale, we maximize the density at the target location point while constraining a specified measure of scale. We consider various choices of scale constraint, such as fixing the concentration parameter, the mean cosine error, or the variance in the Beta case. In several examples, we show that this maximum density method provides superior performance for constructing priors, defining Metropolis-Hastings proposals, and generating simulated probability vectors.