This article considers Bayesian methods for density regression, allowing a random probability distribution to change flexibly with multiple predictors. The conditional response distribution is expressed as a nonparametric mixture of regression models, with the mixture distribution changing with predictors. A class of weighted mixture of Dirichlet process (WMDP) priors is proposed for the uncountable collection of mixture distributions. It is shown that this specification results in a generalized P\'olya urn scheme, which incorporates weights dependent on the distance between subjects' predictor values. To allow local dependency in the mixture distributions, we propose a kernel-based weighting scheme. A Gibbs sampling algorithm is developed for posterior computation. The methods are illustrated using simulated data examples and an epidemiologic application.

Keywords: Conditional density function; Dirichlet process; Local smoothing; Mixture model; Nonparametric Bayes; Generalized P\'olya urn

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