Adaptive Bayesian estimation of conditional discrete-continuous distributions with an application to stock market trading activity

Abstract

We consider Bayesian nonparametric estimation of conditional discrete-continuous distributions. Our model is based on a mixture of normal distributions with covariate dependent mixing probabilities. We use continuous latent variables for modeling the discrete part of the distribution. The marginal distribution of covariates is not modeled. Under anisotropic smoothness conditions on the data generating conditional distribution and a possibly increasing number of the support points for the discrete part of the distribution, we show that the posterior in our model contracts at frequentist adaptive optimal rates up to a log factor. Our results also imply an upper bound on the posterior contraction rate for predictive distributions when the data follow an ergodic Markov process and our model is used for modeling the Markov transition distribution. The proposed model performs well in an application to stock market trading activity.