Weakly Informative Reparameterizations for Location-Scale Mixtures

While mixtures of Gaussian distributions have been studied for more than a century, the construction of a reference Bayesian analysis of those models remains unsolved, with a general prohibition of improper priors due to the ill-posed nature of such statistical objects. This difficulty is usually bypassed by an empirical Bayes resolution. By creating a new parameterization centered on the mean and possibly the variance of the mixture distribution itself, we manage to develop here a weakly informative prior for a wide class of mixtures with an arbitrary number of components. We demonstrate that some posterior distributions associated with this prior and a minimal sample size are proper. We provide Markov chain Monte Carlo (MCMC) implementations that exhibit the expected exchangeability. We only study here the univariate case, the extension to multivariate location-scale mixtures being currently under study. An R package called Ultimixt is associated with this article. Supplementary material for this article is available online.

尽管高斯混合分布（mixtures of Gaussian distributions）的相关研究已逾百年，但针对这类模型构建参考贝叶斯分析的工作仍未解决：由于此类统计对象具有不适定性，这类模型通常无法使用非正常先验分布。这类难题通常可通过经验贝叶斯方法绕过。本文以混合分布自身的均值（乃至方差）为核心构建全新参数化方案，由此为任意分量数的广泛类别的混合模型开发出弱信息先验分布。本文证明，当搭配该先验分布与最小样本量时，部分后验分布为正常后验分布。本文提供了符合预期可交换性的马尔可夫链蒙特卡洛（Markov chain Monte Carlo, MCMC）算法实现。本文仅探讨单变量情形，针对多变量位置-尺度混合模型的拓展研究目前正在进行中。本文配套了一款名为Ultimixt的R软件包，本文的补充材料可在线获取。