  Our article considers a Gaussian variational approximation of the posterior
density in a high-dimensional state space model. The variational parameters to
be optimized are the mean vector and the covariance matrix of the
approximation. The number of parameters in the covariance matrix grows as the
square of the number of model parameters, so it is necessary to find simple yet
effective parameterizations of the covariance structure when the number of
model parameters is large. We approximate the joint posterior distribution over
the high-dimensional state vectors by a dynamic factor model, having Markovian
time dependence and a factor covariance structure for the states. This gives a
reduced description of the dependence structure for the states, as well as a
temporal conditional independence structure similar to that in the true
posterior. The usefulness of the approach is illustrated for prediction in two
high-dimensional applications that are challenging for Markov chain Monte Carlo
sampling. The first is a spatio-temporal model for the spread of the Eurasian
Collared-Dove across North America; the second is a Wishart-based multivariate
stochastic volatility model for financial returns.
