University of Missouri, Columbia, MO, USA.
University of California, Berkeley, Berkeley, CA, USA.
Psychometrika. 2019 Sep;84(3):802-829. doi: 10.1007/s11336-019-09679-0. Epub 2019 Jul 11.
Typical Bayesian methods for models with latent variables (or random effects) involve directly sampling the latent variables along with the model parameters. In high-level software code for model definitions (using, e.g., BUGS, JAGS, Stan), the likelihood is therefore specified as conditional on the latent variables. This can lead researchers to perform model comparisons via conditional likelihoods, where the latent variables are considered model parameters. In other settings, however, typical model comparisons involve marginal likelihoods where the latent variables are integrated out. This distinction is often overlooked despite the fact that it can have a large impact on the comparisons of interest. In this paper, we clarify and illustrate these issues, focusing on the comparison of conditional and marginal Deviance Information Criteria (DICs) and Watanabe-Akaike Information Criteria (WAICs) in psychometric modeling. The conditional/marginal distinction corresponds to whether the model should be predictive for the clusters that are in the data or for new clusters (where "clusters" typically correspond to higher-level units like people or schools). Correspondingly, we show that marginal WAIC corresponds to leave-one-cluster out cross-validation, whereas conditional WAIC corresponds to leave-one-unit out. These results lead to recommendations on the general application of the criteria to models with latent variables.
对于具有潜在变量（或随机效应）的模型，典型的贝叶斯方法涉及直接对潜在变量和模型参数进行抽样。在用于模型定义的高级别软件代码中（例如，使用 BUGS、JAGS、Stan 等），似然因此被指定为条件于潜在变量。这可能导致研究人员通过条件似然来进行模型比较，其中潜在变量被视为模型参数。然而，在其他情况下，典型的模型比较涉及边缘似然，其中潜在变量被积分出来。尽管这可能对感兴趣的比较产生重大影响，但这种区别常常被忽视。在本文中，我们澄清并说明了这些问题，重点是在心理测量建模中比较条件和边缘偏差信息准则 (DIC) 和渡边-赤池信息量准则 (WAIC)。条件/边缘的区别对应于模型应该对数据中的聚类还是新聚类进行预测（其中“聚类”通常对应于更高层次的单位，如人或学校）。相应地，我们表明边缘 WAIC 对应于留一聚类交叉验证，而条件 WAIC 对应于留一单位。这些结果导致了对具有潜在变量的模型的一般应用准则的建议。
