On Robust Probabilistic Principal Component Analysis using Multivariate $t$-Distributions

Principal Component Analysis (PCA) is a common multivariate statistical analysis method, and Probabilistic Principal Component Analysis (PPCA) is its probabilistic reformulation under the framework of Gaussian latent variable model. To improve the robustness of PPCA, it has been proposed to change the underlying Gaussian distributions to multivariate $t$-distributions. Based on the representation of $t$-distribution as a scale mixture of Gaussians, a hierarchical model is used for implementation. However, although the robust PPCA methods work reasonably well for some simulation studies and real data, the hierarchical model implemented does not yield the equivalent interpretation. In this paper, we present a set of equivalent relationships between those models, and discuss the performance of robust PPCA methods using different multivariate $t$-distributed structures through several simulation studies. In doing so, we clarify a current misrepresentation in the literature, and make connections between a set of hierarchical models for robust PPCA.