Bayesian multivariate mixed-scale density estimation

Although univariate continuous density estimation has received abundant attention in the Bayesian nonparametrics literature, there is essentially no theory on multivariate mixed scale density estimation. In this article, we consider a general framework to jointly model continuous, count and categorical variables under a nonparametric prior, which is induced through rounding latent variables having an unknown density with respect to Lesbesgue measure. For the proposed class of priors, we provide sufficient conditions for large support, strong consistency and rates of posterior contraction. These conditions, which primarily relate to the prior on the latent variable density and heaviness of the tails for the observed continuous variables, allow one to convert sufficient conditions obtained in the setting of multivariate continuous density estimation to the mixed scale case. We provide new results in the multivariate continuous density estimation case, showing the Kullback-Leibler property and strong consistency for different mixture priors including priors that parsimoniously model the covariance in a multivariate Gaussian mixture via a sparse factor model. In particular, the results hold for Dirichlet process location and location-scale mixtures of multivariate Gaussians with various prior specifications on the covariance matrix.