Factor models are widely used for dimension reduction in the analysis of multivariate data. This is achieved through decomposition of a p x p covariance matrix into the sum of two components. Through a latent factor representation, they can be interpreted as a diagonal matrix of idiosyncratic variances and a shared variation matrix, that is, the product of a p x k factor loadings matrix and its transpose. If k << p, this defines a parsimonious factorisation of the covariance matrix. Historically, little attention has been paid to incorporating prior information in Bayesian analyses using factor models where, at best, the prior for the factor loadings is order invariant. In this work, a class of structured priors is developed that can encode ideas of dependence structure about the shared variation matrix. The construction allows data-informed shrinkage towards sensible parametric structures while also facilitating inference over the number of factors. Using an unconstrained reparameterisation of stationary vector autoregressions, the methodology is extended to stationary dynamic factor models. For computational inference, parameter-expanded Markov chain Monte Carlo samplers are proposed, including an efficient adaptive Gibbs sampler. Two substantive applications showcase the scope of the methodology and its inferential benefits.

翻译：因子模型在多元数据分析中被广泛用于降维。这是通过将一个p×p协方差矩阵分解为两个分量之和来实现的。通过潜因子表示，它们可以解释为一个由特质方差构成的对角矩阵和一个共享变异矩阵，后者即一个p×k因子载荷矩阵与其转置的乘积。若k << p，这定义了一种简约的协方差矩阵分解方式。历史上，在贝叶斯分析中使用因子模型时，很少关注如何纳入先验信息；其中最好的情况也仅是对因子载荷采用顺序不变的先验。本研究发展了一类结构化先验，能够对共享变异矩阵的依赖结构信息进行编码。该构造允许数据驱动的收缩朝向合理的参数结构，同时也有助于对因子数量进行推断。通过使用平稳向量自回归的无约束重新参数化，该方法被扩展至平稳动态因子模型。为进行计算推断，提出了参数扩展的马尔可夫链蒙特卡洛采样器，包括一种高效的自适应吉布斯采样器。两个实质性应用展示了该方法论的适用范围及其推断优势。