Model Averaging and Dimension Selection for the Singular Value Decomposition

Many multivariate data analysis techniques for an m n matrix Y are related to the model Y = M + E, where Y is an m n matrix of full rank and M is an unobserved mean matrix of rank K< (m n). Typically the rank of M is estimated in a heuristic way and then the least-squares estimate of M is obtained via the singular value decomposition of Y, yielding an estimate that can have a very high variance. In this paper we suggest a model-based alternative to the above approach by providing prior distributions and posterior estimation for the rank of M and the components of its singular value decomposition. In addition to providing more accurate inference, such an approach has the advantage of being extendable to more general data-analysis situations, such as inference in the presence of missing data and estimation in a generalized linear modeling framework.