We all know how to set up Markov chains for MCMC simulation. But, what should we do with the output? The JCGS paper describes how to use the output of an MCMC simulation in order to more effectively perform data analysis--the process of looking at and refining the model by varying prior and likelihood, by varying the structure of how components of the prior fit together, and by varying the data set through case deletion. The method, well beyond standard importance sampling, works extremely well, to the point of potentially potentially rendering inestimable quantities estimable. It applies not only to MCMC methods but also to straight importance sampling methods. The Case Studies paper develops a tool for Bayesian exploratory data analysis. The latter two papers investigate the issue of whether to subsample the output of an MCMC simulation. The 1994 paper shows that the simplest subsampling method decreases the accuracy of estimators. The 2000 paper shows how to increase the accuracy of estimators by subsampling! Although the focus of both papers is on Gibbs sampling, the results apply to more general MCMC algorithms.

• MacEachern, S.N. and M. Peruggia. Bayesian Tools for EDA and Model Building: A Brainy Study. To appear in ``Case Studies in Bayesian Statistics Workshop 5''.
• MacEachern, S.N. and M. Peruggia (2000). Importance Link Function Estimation for Markov Chain Monte Carlo Methods, Journal of Computational and Graphical Statistics 9, 99-121.
• MacEachern, S.N. and M. Peruggia (2000). Subsampling the Gibbs sampler: variance reduction. Statistics & Probability Letters 47, 91-98.
• MacEachern, S.N. and L.M. Berliner (1994). Subsampling the Gibbs Sampler, American Statistician 48, 188-190.

A constant concern with MCMC methods is how quickly they converge to the posterior and how quickly they mix over the posterior. The Communications paper advocates collapsing the state space of the Gibbs sampler in order to improve convergence and provides some supporting theory. The Biometrika paper provides an early example of a "group move". The JCGS paper describes an algorithm where a nonidentifiable model is deliberately created in order to improve the convergence/mixing of the Markov chain used for the simulation. The two book chapters contain additional material. In particular, the "Practical Nonparametrics" chapter has a brief treatment of how to use the notion of piecewise-log-concavity to turn a non-conjugate problem into a conditionally conjugate problem.

• MacEachern, S.N. (1994). Estimating Normal Means with a Conjugate Style Dirichlet Process Prior, Communications in Statistics: Simulation and Computation 23, 727-741.
• Bush, C.A. and S.N. MacEachern (1996). A Semi-parametric Bayesian Model for Randomized Block Designs. Biometrika 83, 275-286.
• MacEachern, S.N. and P. Muller (1998). Estimating Mixture of Dirichlet Process Models. Journal of Computational and Graphical Statistics 7, 223-238.
• MacEachern, S.N. (1998). Computational Methods for Mixture of Dirichlet Process Models, in "Practical Nonparametric and Semiparametric Bayesian Statistics", eds. D. Dey, P. Muller, D. Sinha, 23-44, Springer-Verlag.
• MacEachern, S.N. and Muller, P. Efficient MCMC Schemes for Robust Model Extensions using Encompassing Dirichlet Process Mixture Models in "Robust Bayesian Analysis", eds. D. Rios Insua and F. Ruggeri, 295-315, Springer-Verlag.

Round-off error has been a bugaboo for computer scientists since the development of the computer. They have developed the technique of interval analysis which continually tracks upper and lower bounds for unrounded versions of a calculation. In the paper below, we exploit this technique to track the support of chaotic, dynamical systems corrupted by error.

• Berliner, L.M., S.N. MacEachern, and C.M. Scipione (1997). Ergodic Distributions of Random Dynamical Systems. Fields Institute Communications 11, 171-185.