Variational inference is an alternative estimation technique for Bayesian models. Recent work shows that variational methods provide consistent estimation via efficient, deterministic algorithms. Other tools, such as model selection using variational AICs (VAIC) have been developed and studied for the linear regression case. While mixed effects models have enjoyed some study in the variational context, tools for model selection are lacking. One important feature of model selection in mixed effects models, particularly longitudinal models, is the selection of the random effects which in turn determine the covariance structure for the repeatedly sampled outcome. To address this, we derive a VAIC specifically for variational mixed effects (VME) models. We also implement a parameter-efficient VME as part of our study which reduces any general random effects structure down to a single subject-specific score. This model accommodates a wide range of random effect structures including random intercept and slope models as well as random functional effects. Our VAIC can model and perform selection on a variety of VME models including more classic longitudinal models as well as longitudinal scalar-on-function regression. As we demonstrate empirically, our VAIC performs well in discriminating between correctly and incorrectly specified random effects structures. Finally, we illustrate the use of VAICs for VMEs on two datasets: a study of lead levels in children and a study of diffusion tensor imaging.

翻译：变分推断是贝叶斯模型的一种替代估计技术。近期研究表明，变分方法通过高效的确定性算法提供一致性估计。其他工具，如基于变分AIC（VAIC）的模型选择，已在线性回归案例中得到开发与研究。尽管混合效应模型在变分框架下已有一些研究，但模型选择工具仍显不足。混合效应模型（尤其是纵向模型）中模型选择的重要特征之一是对随机效应的选择，这决定了重复测量结果的协方差结构。为此，我们专门针对变分混合效应（VME）模型推导出VAIC。作为研究的一部分，我们还实现了一种参数高效的VME，将一般随机效应结构简化为单一的受试者特异性评分。该模型可容纳广泛的随机效应结构，包括随机截距模型、随机斜率模型以及随机函数效应模型。我们的VAIC能够对各种VME模型进行建模与选择，包括经典纵向模型及纵向标量-函数回归模型。实证研究表明，我们的VAIC在区分正确与错误设定的随机效应结构方面表现良好。最后，我们通过两个数据集——儿童血铅水平研究以及弥散张量成像研究——展示了VAIC用于VME的应用实例。