Tuning tempered transitions

The method of tempered transitions was proposed by Neal (Stat. Comput. 6:353-366, 1996) for tackling the difficulties arising when using Markov chain Monte Carlo to sample from multimodal distributions. In common with methods such as simulated tempering and Metropolis-coupled MCMC, the key idea is to utilise a series of successively easier to sample distributions to improve movement around the state space. Tempered transitions does this by incorporating moves through these less modal distributions into the MCMC proposals. Unfortunately the improved movement between modes comes at a high computational cost with a low acceptance rate of expensive proposals. We consider how the algorithm may be tuned to increase the acceptance rates for a given number of temperatures. We find that the commonly assumed geometric spacing of temperatures is reasonable in many but not all applications.