  We implement gradient-based variational inference routines for Wishart and
inverse Wishart processes, which we apply as Bayesian models for the dynamic,
heteroskedastic covariance matrix of a multivariate time series. The Wishart
and inverse Wishart processes are constructed from i.i.d. Gaussian processes,
existing variational inference algorithms for which form the basis of our
approach. These methods are easy to implement as a black-box and scale
favorably with the length of the time series, however, they fail in the case of
the Wishart process, an issue we resolve with a simple modification into an
additive white noise parameterization of the model. This modification is also
key to implementing a factored variant of the construction, allowing inference
to additionally scale to high-dimensional covariance matrices. Through
experimentation, we demonstrate that some (but not all) model variants
outperform multivariate GARCH when forecasting the covariances of returns on
financial instruments.
