## 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.