Authors’ rejoinder to Markov chain Monte Carlo: Some practical implications of theoretical results

1. Dirichlet process priors. Ishwaran has presented an analysis of a model (the Rasch model) using MCMC with Dirichlet process priors, following the approach of Escobar (1994) and MacEachern (1994). The analysis is a good example of how applied statisticians should approach such problems. Ishwaran is well informed about available theoretical results, and makes use of them where possible. At the same time, he is aware of the current limitations of such results, and is not afraid to use ad-hoc techniques and intuitive reasoning where necessary. We wish that more applied users would strike this same balance. Ishwaran concludes his analysis by discussing how to monitor the convergence of his MCMC algorithm. We certainly agree with him when he says “The ideal solution would have been to derive quantitative rates of convergence”. Coincidentally, we have recently begun working (Petrone, Roberts, and Rosenthal, 1997) on convergence results for models similar to that which he considers. However, this work has encountered many obstacles, and currently available results are not good enough to be of practical value. This highlights the need for a pragmatic approach to implementation, as both discussants have argued. Because of these difficulties, it is perfectly reasonable that Ishwaran instead uses convergence diagnostics, following Gelman and Rubin (1992), a popular approach (see Brooks