Bayesian analysis of reduced rank regression

Abstract

Multivariate regression is a well known method to relate multivariate responses to multivariate predictors. When the responses follow a factor analysis model and the latent factors can be linked with the predictors, a reduced rank regression model is obtained which is defined as a multivariate regression model with rank restrictions on the coefficient matrix. Many linear multivariate methods can be interpreted as special cases of such a reduced rank regression model. A Bayesian analysis with informative conjugate priors is presented, emphasizing the importance of an adequate parametrization to obtain interpretable results. Gibbs sampling a Markov chain Monte Carlo method, is proposed to calculate the posterior and predictive distribution. The methods are motivated and illustrated by an example where quantitative relationships between biological activity and chemical structure are searched for (QSAR).