Mixtures of linear mixed models are widely used for modelling longitudinal data for which observation times differ between subjects. In typical applications, temporal trends are described using a basis expansion, with basis coefficients treated as random effects varying by subject. Additional random effects can describe variation between mixture components, or other known sources of variation in complex experimental designs. A key advantage of these models is that they provide a natural mechanism for clustering, which can be helpful for interpretation in many applications. Current versions of mixtures of linear mixed models are not specifically designed for the case where there are many observations per subject and a complex temporal trend, which requires a large number of basis functions to capture. In this case, the subject-specific basis coefficients are a high-dimensional random effects vector, for which the covariance matrix is hard to specify and estimate, especially if it varies between mixture components. To address this issue, we consider the use of recently-developed deep mixture of factor analyzers models as the prior for the random effects. The resulting deep mixture of linear mixed models is well-suited to high-dimensional settings, and we describe an efficient variational inference approach to posterior computation. The efficacy of the method is demonstrated on both real and simulated data.

翻译：线性混合模型混合广泛应用于建模观察时间因研究对象而异的纵向数据。在典型应用中，时间趋势通过基函数展开来描述，基系数被视为随研究对象变化的随机效应。额外的随机效应可描述混合成分之间的变异，或复杂实验设计中其他已知变异来源。这些模型的一个关键优势在于，它们提供了自然的聚类机制，有助于许多应用场景中的解释性分析。当前版本的线性混合模型混合并未专门设计用于处理每个研究对象有大量观测值且存在复杂时间趋势的情况——后者需要大量基函数才能捕捉。在这种情形下，研究对象特定的基系数构成了高维随机效应向量，其协方差矩阵难以指定和估计，尤其当协方差矩阵在不同混合成分间存在差异时。为解决该问题，我们考虑将近期发展的深度混合因子分析器模型作为随机效应的先验分布。由此产生的深度混合线性混合模型非常适合高维场景，并描述了一种高效的后验计算变分推断方法。通过在真实数据与模拟数据上的实验，验证了该方法的有效性。