  Probabilistic principal component analysis (PPCA) is a probabilistic
reformulation of principal component analysis (PCA), under the framework of a
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 Gaussian distributions, a hierarchical model is used for
implementation. However, in the existing literature, the hierarchical model
implemented does not yield the equivalent interpretation.
  In this paper, we present two sets of equivalent relationships between the
high-level multivariate $t$-PPCA framework and the hierarchical model used for
implementation. In doing so, we clarify a current misrepresentation in the
literature, by specifying the correct correspondence. In addition, we discuss
the performance of different multivariate $t$ robust PPCA methods both in
theory and simulation studies, and propose a novel Monte Carlo
expectation-maximization (MCEM) algorithm to implement one general type of such
models.
