A Fully Nonparametric Modelling Approach to Binary Regression

We propose a general nonparametric Bayesian framework for binary regression, which is built from modelling for the joint response-covariate distribution. The observed binary responses are assumed to arise from underlying continuous random variables through discretization, and we model the joint distribution of these latent responses and the covariates using a Dirichlet pro-cess mixture of multivariate normals. We show that the kernel of the induced mixture model for the observed data is identifiable upon a restriction on the latent variables. To allow for appro-priate dependence structure while facilitating identifiability, we use a square-root-free Cholesky decomposition of the covariance matrix in the normal mixture kernel. In addition to allowing for the necessary restriction, this modelling strategy provides substantial simplifications in im-plementation of Markov chain Monte Carlo posterior simulation. We illustrate the utility of the modelling approach with two data examples, and discuss extensions to incorporate multivariate ordinal responses, as well as mixed ordinal-continuous responses