Generalized linear mixed models (GLMMs) are commonly used to analyze correlated discrete or continuous response data. In Bayesian GLMMs, the often-used improper priors may yield undesirable improper posterior distributions. Thus, verifying posterior propriety is crucial for valid applications of Bayesian GLMMs with improper priors. Here, we consider the popular improper uniform prior on the regression coefficients and several proper or improper priors, including the widely used gamma and power priors on the variance components of the random effects. We also construct an approximate Jeffreys' prior for objective Bayesian analysis of GLMMs. We derive necessary and sufficient conditions for posterior propriety for Bayesian GLMMs where the response variables have distributions from the exponential family. For the two most widely used GLMMs, namely, the binomial and Poisson GLMMs, we further refine our results by providing easily verifiable conditions compared to the currently available results. Finally, we use examples involving one-way and two-way random effects models to demonstrate the theoretical results derived here.

翻译：广义线性混合模型（GLMMs）常用于分析相关的离散或连续响应数据。在贝叶斯GLMMs中，常用的非正常先验可能导致不良的非正常后验分布。因此，验证后验适定性对于确保含非正常先验的贝叶斯GLMMs有效应用至关重要。本文考虑了回归系数上常用的非正常均匀先验，以及随机效应方差分量上若干正常或非正常先验，包括广泛使用的伽马先验和幂先验。我们还为客观贝叶斯分析GLMMs构造了近似Jeffreys先验。针对响应变量服从指数族分布的贝叶斯GLMMs，我们推导了后验适定性的充要条件。对于最常用的两种GLMMs，即二项和泊松GLMMs，我们进一步优化了结果，提供了比现有结果更易验证的条件。最后，通过单向和双向随机效应模型的实例，验证了本文推导的理论结果。