In previous chapters, we have presented the theoretical foundations of MCMC algorithms and showed that, under fairly general conditions, the chains produced by these algorithms are ergodic, or even geometrically ergodic. While such developments are obviously necessary, they are nonetheless insufficient from the point of view of the implementation of MCMC methods. They do not directly result in methods of controlling the chain produced by an algorithm (in the sense of a stopping rule to guarantee that the number of iterations is sufficient). In other words, while necessary as mathematical proofs of the validity of the MCMC algorithms, general convergence results do not tell us when to stop these algorithms and produce our estimates. For instance, the mixture model of Example 10.18 is fairly well behaved from a theoretical point of view, but Figure 10.3 indicates that the number of iterations used is definitely insufficient.
KeywordsMarkov Chain Stationary Distribution Gibbs Sampler Importance Sampling Transition Kernel
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