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 sparse 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：因子模型广泛应用于多元数据分析中的降维。这通过将p×p协方差矩阵分解为两个分量之和来实现。通过潜变量因子表示，它们可以被解释为特异方差的对角矩阵和共享变异矩阵（即p×k因子载荷矩阵与其转置的乘积）。若k << p，则定义了对协方差矩阵的稀疏分解。历史上，在贝叶斯分析中利用因子模型融入先验信息时，对因子载荷的（至多）顺序不变性先验关注甚少。本研究开发了一类结构化先验，能够编码关于共享变异矩阵的依赖结构信息。该构建方法允许数据驱动地向合理的参数结构收缩，同时支持对因子数量的推断。通过使用平稳向量自回归的无约束重参数化，该方法被扩展至平稳动态因子模型。在计算推断方面，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。两个实质性应用案例展示了该方法的适用范围及其推断优势。