Partition Weighted Approach for Estimating the Marginal Posterior Density with Applications

The computation of marginal posterior density in Bayesian analysis is essential in that it can provide complete information about parameters of interest. Furthermore, the marginal posterior density can be used for computing Bayes factors, posterior model probabilities, and diagnostic measures. The conditional marginal density estimator (CMDE) is theoretically the best for marginal density estimation but requires the closed-form expression of the conditional posterior density, which is often not available in many applications. We develop the partition weighted marginal density estimator (PWMDE) to realize the CMDE. This unbiased estimator requires only a single MCMC output from the joint posterior distribution and the known unnormalized posterior density. The theoretical properties and various applications of the PWMDE are examined in detail. The PWMDE method is also extended to the estimation of conditional posterior densities. We carry out simulation studies to investigate the empirical performance of the PWMDE and further demonstrate the desirable features of the proposed method with two real data sets from a study of dissociative identity disorder patients and a prostate cancer study, respectively.

在贝叶斯分析中，边缘后验密度的计算至关重要，因为它能够提供关于感兴趣参数的完整信息。此外，边缘后验密度还可用于计算贝叶斯因子、后验模型概率和诊断指标。条件边缘密度估计器（CMDE）在理论上是最优的边缘密度估计方法，但其要求条件后验密度的封闭形式表达式，这在许多应用中往往难以获得。本研究开发了一种分区加权边缘密度估计器（PWMDE），以实现CMDE。该无偏估计器仅需从联合后验分布中提取单个MCMC输出以及已知的未归一化后验密度。PWMDE的理论性质及其多种应用被详细考察。PWMDE方法也被扩展至条件后验密度的估计。本研究通过模拟研究探讨了PWMDE的实证性能，并利用来自分离性身份障碍患者研究和前列腺癌研究的两个真实数据集，进一步展示了所提出方法的优势特征。