Bayesian Mixture Modeling for Multivariate Conditional Distributions

We present a Bayesian mixture model for estimating the joint distribution of mixed ordinal, nominal, and continuous data conditional on a set of fixed variables. The modeling strategy is motivated by applied contexts in marketing and the social sciences, in particular data fusion and the analysis of stratified or quota samples. The model uses multivariate normal and categorical mixture kernels for the random variables. It induces dependence between the random and fixed variables through the means of the multivariate normal mixture kernels and via a truncated local Dirichlet process. The latter encourages observations with similar values of the fixed variables to share mixture components. We illustrate use of the model for missing data imputation, in particular data fusion of two surveys, and for the analysis of stratified or quota samples. The data fusion example suggests that the model can estimate underlying relationships in the data and the distributions of the missing values more accurately than several other approaches, including a mixture model applied to the random and fixed variables jointly. We also use the model to analyze consumers' reading behaviors from a quota sample, i.e., a sample where the empirical distribution of some variables is fixed by design and so should not be modeled as random, conducted by the book publisher HarperCollins.