Bayesian analyses of multivariate binary or categorical outcomes typically rely on probit or mixed effects logistic regression models which do not have a marginal logistic structure for the individual outcomes. In addition, difficulties arise when simple non-informative priors are chosen for the covariance parameters. Motivated by these problems, we propose a new type of multivariate logistic distribution that can be used to construct a likelihood for multivariate logistic regression analysis of binary and categorical data. The model for individual outcomes has a marginal logistic structure, simplifying interpretation. We follow a Bayesian approach to estimation and inference, developing an efficient data augmentation algorithm for posterior computation. Propriety results are provided under uniform improper priors, and the method is illustrated with application to a study of fetal growth restriction in twins.

Keywords: Block updating; categorical data; data augmentation; latent variables; MCMC algorithm; multiple binary outcomes; proportional odds