As mentioned in Chapter 9 section 5.1 (yyy version 9/1/99) there has been intense theoretical study of the problem of sampling uniformly from a convex set in , in the limit. This problem turns out to be essentially equivalent to the problem of sampling from a log-concave density , that is a density of the form for convex . The results are not easy to state; see Bubley et al [78] for discussion.
Here is a special setting in which one can make rigorous inferences from MCMC without rigorous bounds on mixing times. Suppose we have a guess at the relaxation time of a Markov sampler from a traget distribution ; suppose we have some separate method of sampling exactly from , but where the cost of one exact sample is larger than the cost of steps of the Markov sampler. In this setting it is natural to take exact samples and use them as initial states of multiple runs of the Markov sampler. It turns out (see [5] for precise statement) that one can obtain confidence intervals for a mean which are always rigorously correct (without assumptions on ) and which, if is indeed approximately , will have optimal length, that is the length which would be implied by this value of .