Lack of consistency of mean field and variational Bayes approximations for state space models

The consistency problem of both mean field and variational Bayes estimators in the context of linear state space models is investigated. We prove that the mean field approximation is asymptotically consistent when the variances of the noise variables in the system are sufficiently small, but neither the mean field estimator nor the variational Bayes estimator is always asymptotically consistent as the lsquosample sizersquo becomes large. The lsquogaprsquo between the estimators and the true values is roughly estimated.