Bayesian Covariance Regression and Autoregression

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. A number of multivariate heteroscedastic time series models have been proposed within the econometrics literature, but are typically limited by lack of clear margins, computational intractability, and curse of dimensionality. In this talk, we first introduce and explore a new class of time series models for covariance matrices based on a constructive definition exploiting inverse Wishart distribution theory. The construction yields a stationary, first-order autoregressive (AR) process on the cone of positive semi-definite matrices.

We then turn our focus to more general predictor spaces and scaling to high-dimensional datasets. Our proposed Bayesian nonparametric covariance regression framework harnesses a latent factor model representation. In particular, the predictor-dependent factor loadings are characterized as a sparse combination of a collection of unknown dictionary functions (e.g, Gaussian process random functions). The induced predictor-dependent 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 an application to the Google Flu Trends data and the task of word classification based on single-trial MEG data.

Joint work with David Dunson and Mike West.