Sampling from Multimodal Distributions using Tempered Transitions

Radford M. Neal, Dept. of Statistics and Dept. of Computer Science, University of Toronto

I present a new Markov chain sampling method appropriate for distributions with isolated modes. Like the recently-developed method of ``simulated tempering'', the ``tempered transition'' method uses a series of distributions that interpolate between the distribution of interest and a distribution for which sampling is easier. The new method has the advantage that it does not require approximate values for the normalizing constants of these distributions, which are needed for simulated tempering, and can be tedious to estimate. Simulated tempering performs a random walk along the series of distributions used. In contrast, the tempered transitions of the new method move systematically from the desired distribution, to the easily-sampled distribution, and back to the desired distribution. This systematic movement avoids the inefficiency of a random walk, an advantage that unfortunately is cancelled by an increase in the number of interpolating distributions required. Because of this, the sampling efficiency of the tempered transition method in simple problems is similar to that of simulated tempering. On more complex distributions, however, simulated tempering and tempered transitions may perform differently. Which is better depends on the ways in which the interpolating distributions are ``deceptive''.

Technical Report No. 9421, Dept. of Statistics (October 1994), 22 pages: postscript, pdf.

Associated reference: A revised version of this paper has now been published:
Neal, R. M. (1996) ``Sampling from multimodal distributions using tempered transitions'' Statistics and Computing, vol. 6, pp. 353-366: abstract.

The following technical report describes a method related to tempered transitions:

Neal, R. M. (1998) ``Annealed importance sampling'', Technical Report No. 9805 (revised), Dept. of Statistics, University of Toronto, 25 pages: abstract, postscript, pdf.