Abstract

We derive streamlined mean field variational Bayes algorithms for fitting linear mixed models with crossed random effects. In the most general situation, where the dimensions of the crossed groups are arbitrarily large, streamlining is hindered by lack of sparseness in the underlying least squares system. Because of this fact we also consider a hierarchy of relaxations of the mean field product restriction. The least stringent product restriction delivers a high degree of inferential accuracy. However, this accuracy must be mitigated against its higher storage and computing demands. Faster sparse storage and computing alternatives are also provided, but come with the price of diminished inferential accuracy. This article provides full algorithmic details of three variational inference strategies, presents detailed empirical results on their pros and cons and, thus, guides the users on their choice of variational inference approach depending on the problem size and computing resources. Supplementary materials for this article are available online.

 抽象的

我们推导了简化的平均场变分贝叶斯算法，用于拟合具有交叉随机效应的线性混合模型。在最一般的情况下，交叉组的维度任意大，流线化会因底层最小二乘系统缺乏稀疏性而受到阻碍。因此，我们还考虑了平均场积限制的放宽层次。最不严格的产品限制可提供高度的推理准确性。然而，这种准确性必须根据其更高的存储和计算需求而降低。还提供了更快的稀疏存储和计算替代方案，但代价是推理准确性降低。本文提供了三种变分推理策略的完整算法细节，并就其优缺点提供了详细的实证结果，从而指导用户根据问题规模和计算资源选择变分推理方法。本文的补充材料可在线获取。