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Abstract
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.