MCMC algorithms for constrained variance matrices

The problem of finding a generic algorithm for applying Markov chain Monte Carlo (MCMC) estimation procedures to statistical models that include variance matrices with additional parameter constraints is considered. Such problems can be split between additional constraints across variance matrices and within variance matrices. The case of additional constraints across variance matrices is considered here for the first time and a review of existing work on the case of additional parameter constraints within a variance matrix is given. Two simple single-site updating random walk Metropolis algorithms are described which have the advantage of generality in that they can be applied to virtually all scenarios. Four applications where these methods can be used in practice are given. Some situations when such single-site algorithms break down are described and multiple-site alternatives are briefly discussed.