We propose a general nonparametric Bayesian framework for binary regression,which is built from modeling for the joint response-covariate distribution. Theobserved binary responses are assumed to arise from underlying continuousrandom variables through discretization, and we model the joint distribution ofthese latent responses and the covariates using a Dirichlet process mixture ofmultivariate normals. We show that the kernel of the induced mixture model forthe observed data is identifiable upon a restriction on the latent variables.To allow for appropriate dependence structure while facilitatingidentifiability, we use a square-root-free Cholesky decomposition of thecovariance matrix in the normal mixture kernel. In addition to allowing for thenecessary restriction, this modeling strategy provides substantialsimplifications in implementation of Markov chain Monte Carlo posteriorsimulation. We present two data examples taken from areas for which themethodology is especially well suited. In particular, the first exampleinvolves estimation of relationships between environmental variables, and thesecond develops inference for natural selection surfaces in evolutionarybiology. Finally, we discuss extensions to regression settings withmultivariate ordinal responses.
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