Annealing Markov Chain Monte Carlo with Applications to Ancestral Inference

Abstract Markov chain Monte Carlo (MCMC; the Metropolis-Hastings algorithm) has been used for many statistical problems, including Bayesian inference, likelihood inference, and tests of significance. Though the method generally works well, doubts about convergence often remain. Here we propose MCMC methods distantly related to simulated annealing. Our samplers mix rapidly enough to be usable for problems in which other methods would require eons of computing time. They simulate realizations from a sequence of distributions, allowing the distribution being simulated to vary randomly over time. If the sequence of distributions is well chosen, then the sampler will mix well and produce accurate answers for all the distributions. Even when there is only one distribution of interest, these annealing-like samplers may be the only known way to get a rapidly mixing sampler. These methods are essential for attacking very hard problems, which arise in areas such as statistical genetics. We illustrate the methods with an application that is much harder than any problem previously done by MCMC, involving ancestral inference on a very large genealogy (7 generations, 2,024 individuals). The problem is to find, conditional on data on living individuals, the probabilities of each individual having been a carrier of cystic fibrosis. Exact calculation of these conditional probabilities is infeasible. Moreover, a Gibbs sampler for the problem would not mix in a reasonable time, even on the fastest imaginable computers. Our annealing-like samplers have mixing times of a few hours. We also give examples of samplers for the “witch's hat” distribution and the conditional Strauss process.