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.

翻译：狄利克雷分布常用于建模概率单纯形中的向量。当作为先验分布或建议分布使用时，通常会将狄利克雷分布的均值设定为期望分布中心的位置。然而，若均值靠近概率单纯形的边界，狄利克雷分布将高度集中于以下两种情况之一：(i) 均值点附近，或(ii) 极度接近边界处。因此，以均值为中心会导致边界附近位置与尺度的控制效果不佳。本文提出一种改进Beta分布与狄利克雷分布位置与尺度控制的方法。具体而言，在给定目标位置点与期望尺度的条件下，我们通过约束特定尺度度量来最大化目标位置点处的分布密度。我们考虑了多种尺度约束的选择，例如固定浓度参数、平均余弦误差或Beta分布情形下的方差。通过若干算例，我们证明这种最大密度方法在构建先验分布、定义Metropolis-Hastings建议分布以及生成模拟概率向量方面具有更优的性能。