Empirical Bayes Density Regression

In Bayesian hierarchical modeling, it is often appealing to allow the conditional density of an (observable or unobservable) random variable Y to change flexibly with categorical and continuous predictors X. A mixture of regression models is proposed, with the mixture distribution varying with X. Treating the smoothing parameters and number of mixture components as unknown, the MLE does not exist, motivating an empirical Bayes approach. The proposed method shrinks the spatially-adaptive mixture distributions to a common baseline, while penalizing rapid changes and large numbers of components. The discrete form of the mixture distribution facilitates flexible classification of subjects. A Gibbs sampling algorithm is developed, which embeds a Monte Carlo EM-type stage to estimate smoothing and hyper-parameters. The method is applied to simulated examples and data from an epidemiologic study.