Bayesian multivariate mixed-scale density estimation

TL;DR

This paper develops a Bayesian nonparametric framework for joint density estimation of mixed-scale data, including continuous, count, and categorical variables, with theoretical guarantees and practical application.

Contribution

It introduces a general approach for multivariate mixed-scale density estimation with theoretical support and demonstrates its application on real data.

Findings

Provided conditions for large support and posterior consistency.

Established rates of posterior contraction for the proposed model.

Applied the method successfully to a crime and communities dataset.

Abstract

Although continuous density estimation has received abundant attention in the Bayesian nonparametrics literature, there is limited theory on multivariate mixed scale density estimation. In this note, 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 Lebesgue measure. For the proposed class of priors, we provide sufficient conditions for large support, strong consistency and rates of posterior contraction. These conditions allow one to convert sufficient conditions obtained in the setting of multivariate continuous density estimation to the mixed scale case. To illustrate the procedure a rounded multivariate nonparametric mixture of Gaussians is introduced and applied to a crime and communities dataset.