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：因子模型在多变量数据分析中广泛用于降维，其核心在于将p×p协方差矩阵分解为两个分量之和。通过潜因子表示，该模型可解释为特异方差的对角矩阵与共享变异矩阵（即p×k因子载荷矩阵与其转置的乘积）的叠加。当k<<p时，这定义了协方差矩阵的简约分解。在贝叶斯因子模型分析中，先验信息的融入长期未受重视，至多仅采用顺序不变的因子载荷先验。本研究发展了一类结构化先验分布，能够编码关于共享变异矩阵依赖结构的先验认知。该构造支持数据驱动的收缩至合理参数结构，同时便于对因子数量进行推断。通过引入平稳向量自回归的无约束重参数化，该方法进一步扩展至平稳动态因子模型。为实现计算推断，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。