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 lowdimensional 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 is 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 a priori independent, but exchangeable. Such a hierarchical approach allows more flexibility in the types of patterns that can be represented. Some key words: Bayesian, multiplicative model, PARAFAC, regularization, shrinkag