Neal Madras - In search of faster simulations

Markov Chain Monte Carlo has been one of the most active topics in applied probability in recent years. The main idea is that one can generate random samples from a given complicated distribution by first inventing a Markov chain whose equilibrium distribution is the distribution of interest, and then simulating the chain on a computer. Unfortunately, the chain may converge to equilibrium very slowly; for example, this can be the case when the equilibrium distribution is strongly multimodal. One class of remedies involves creating a sequence of overlapping distributions that interpolate between the distribution of interest and some ``easy'' distribution (whose associated Markov chain equilibriates rapidly). These methods have been successful in practice, but the mathematical theory behind them is not yet complete.