Since the early 1990s, the use of Markov Chain Monte Carlo methods has led to dramatic increases in the use of Bayesian modeling approaches. While this has generally been a good thing, the use of Markov Chains is potentially dangerous as the utilization of these methods without an ability to verify convergence could lead even sophisticated users to report incorrect results based on Markov chains that have not achieved convergence or mixed properly.
My research has been focused in two areas. The first relates to attempts at deriving iid sampling procedures for a number of Bayesian models. For Bayesian generalized linear models, I have had a fair amount of success with the first two papers below. The first paper, which appeared in the Journal of the American Statistical Association, contains an iid sampling procedure for models with multivariate normal priors and log-concave likelihood functions. The procedure appears to be quite fast for models up to a dimension of about 10-12. The second paper also contains effcicient sampling procedures for sampling from models with normal or multivariate normal data. All of these procedure are contained in my R-package Baldur and are being incorporated into my corresponding Excel Add-in. While I have been able to extend some of these results to more complex hierarchical models, as in the third paper, the acceptance rates affiliated with accept-reject procedures for such procedures tend to be low and hence the procedures appear to be impractical.
My second area of research has been in trying to find methods for assessing the convergence properties and rates of convergence for Markov Chains. For two block-gibbs samplers, I have has a fair amount of success as demonstrated in the the fourth paper below.That paper essentially shows that such models under certain conditions exhibit geometrically fast convergence. Interestingly, for some of the models, the requirement for geometric convergence tends to be similar to conditions under which it is possible to construct exact sampling procedures for the same models.
Nygren, K., and Nygren, L. Likelihood Subgradient Densities, Journal of the American Statistical Association, 2006, vol. 101. p.1144-1156.
Nygren, K. Efficient Exact Sampling Procedures for Bayesian Generalized Linear Models. Unpublished Manuscript. 2004.
Nygren, K. Exact Sampling Procedures for Hierarchical Bayesian Models. Unpublished Manuscript. 2005.
Nygren, K. On the Geometric Ergodicity of Two-Block Gibbs Samplers. In Progress.