  Reduced-rank decompositions provide descriptions of the variation among the
elements of a matrix or array. In such decompositions, the elements of an array
are expressed as products of low-dimensional latent factors. This article
presents a model-based version of such a decomposition, extending the scope of
reduced rank methods to accommodate a variety of data types such as
longitudinal social networks and continuous multivariate data that are
cross-classified by categorical variables. The proposed model-based approach is
hierarchical, in that the latent factors corresponding to a given dimension of
the array are not {\it a priori} independent, but exchangeable. Such a
hierarchical approach allows more flexibility in the types of patterns that can
be represented.
