  Although there is a rich literature on methods for allowing the variance in a
univariate regression model to vary with predictors, time and other factors,
relatively little has been done in the multivariate case. Our focus is on
developing a class of nonparametric covariance regression models, which allow
an unknown p x p covariance matrix to change flexibly with predictors. The
proposed modeling framework induces a prior on a collection of covariance
matrices indexed by predictors through priors for predictor-dependent loadings
matrices in a factor model. In particular, the predictor-dependent loadings are
characterized as a sparse combination of a collection of unknown dictionary
functions (e.g, Gaussian process random functions). The induced covariance is
then a regularized quadratic function of these dictionary elements. Our
proposed framework leads to a highly-flexible, but computationally tractable
formulation with simple conjugate posterior updates that can readily handle
missing data. Theoretical properties are discussed and the methods are
illustrated through simulations studies and an application to the Google Flu
Trends data.
