## Modifying Gibbs Sampling to Avoid Self Transitions

** Radford M. Neal,
Dept. of Statistical Sciences, University of Toronto
**
Gibbs sampling is a popular Markov chain Monte Carlo method that
samples from a distribution on *n* state variables by repeatedly
sampling from the conditional distribution of one variable,
*x*_{i}, given the other variables,
*x*_{-i}, either choosing *i* randomly, or updating
sequentially using some systematic or random order for *i*. When
*x*_{i} is discrete, a Gibbs sampling update may choose a new value
that is the same as the old value. A theorem of Peskun indicates
that, when *i* is chosen randomly, a reversible method that
reduces the probability of such self transitions, while increasing the
probabilities of transitioning to each of the other values, will
decrease the asymptotic variance of estimates of expectations of
functions of the state. This has inspired two modified Gibbs sampling
methods, originally due to Frigessi, Hwang, and Younes and to Liu,
though these do not always reduce self transitions to the minimum
possible. Methods that do reduce the probability of self transitions
to the minimum, but do not satisfy the conditions of Peskun's theorem,
have also been devised, by Suwa and Todo, some of which are reversible
and some not. I review and relate these past methods, and introduce a
broader class of reversible methods, including that of Frigessi,
\textit{et al.}, based on what I call ``antithetic modification'',
which also reduce asymptotic variance compared to Gibbs sampling, even
when not satisfying the conditions of Peskun's theorem. A
modification of one method in this class, which I denote as ZDNAM,
reduces self transitions to the minimum possible, while still always
reducing asymptotic variance compared to Gibbs sampling. I introduce
another new class of non-reversible methods based on slice sampling
that can also minimize self transition probabilities. I provide
explicit, efficient implementations of all these methods, and compare
the performance of Gibbs sampling and these modified Gibbs sampling
methods in simulations of a 2D Potts model, a Bayesian mixture model,
and a belief network with unobserved variables. The assessments look
at both random selection of *i*, and several sequential update
schemes. Sequential updates using methods that minimize self
transition probabilities are found to usually be superior, with ZDNAM
often performing best. There is evidence that the non-reversibility
produced by sequential updating can be beneficial, but no consistent
benefit is seen from the individual updates being done by a
non-reversible method.

Technical report, 26 March 2024, 84 pages:
pdf.

Also available at
arXiv.org.

**Associated reference:**
The following is a companion paper that develops some of the theory used:
Neal, R. M. and Rosenthal, J. S. (2023) ``Efficiency of reversible MCMC methods:
Elementary derivations and applications to composite methods'', Technical
Report, 24 pages:
abstract,
pdf.