  Mixtures of factor analysers (MFA) models represent a popular tool for
finding structure in data, particularly high-dimensional data. While in most
applications the number of clusters, and especially the number of latent
factors within clusters, is mostly fixed in advance, in the recent literature
models with automatic inference on both the number of clusters and latent
factors have been introduced. The automatic inference is usually done by
assigning a nonparametric prior and allowing the number of clusters and factors
to potentially go to infinity. The MCMC estimation is performed via an adaptive
algorithm, in which the parameters associated with the redundant factors are
discarded as the chain moves. While this approach has clear advantages, it also
bears some significant drawbacks. Running a separate factor-analytical model
for each cluster involves matrices of changing dimensions, which can make the
model and programming somewhat cumbersome. In addition, discarding the
parameters associated with the redundant factors could lead to a bias in
estimating cluster covariance matrices. At last, identification remains
problematic for infinite factor models. The current work contributes to the MFA
literature by providing for the automatic inference on the number of clusters
and the number of cluster-specific factors while keeping both cluster and
factor dimensions finite. This allows us to avoid many of the aforementioned
drawbacks of the infinite models. For the automatic inference on the cluster
structure, we employ the dynamic mixture of finite mixtures (MFM) model.
Automatic inference on cluster-specific factors is performed by assigning an
exchangeable shrinkage process (ESP) prior to the columns of the factor loading
matrices. The performance of the model is demonstrated on several benchmark
data sets as well as real data applications.
