## Efficiency of Reversible MCMC Methods: Elementary
Derivations and Applications to Composite Methods

** Radford M. Neal
and Jeffrey S. Rosenthal,
Dept. of Statistical Sciences, University of Toronto
**
We review criteria for comparing the efficiency
of Markov chain Monte Carlo (MCMC) methods with respect to the
asymptotic variance of estimates of expectations of functions of
state, and show how such criteria can justify ways of combining
improvements to MCMC methods. We say that a chain on a finite state
space with transition matrix *P* efficiency-dominates one with
transition matrix *Q* if for every function of state it has lower (or
equal) asymptotic variance. We give elementary proofs of some
previous results regarding efficiency dominance, leading to a
self-contained demonstration that a reversible chain with transition
matrix *P* efficiency-dominates a reversible chain with transition
matrix *Q* if and only if none of the eigenvalues of *Q-P* are
negative. This allows us to conclude that modifying a reversible MCMC
method to improve its efficiency will also improve the efficiency of a
method that randomly chooses either this or some other reversible method,
and to conclude that improving the efficiency of a reversible update
for one component of state (as in Gibbs sampling) will improve the
overall efficiency of a reversible method that combines this and other updates.
It also explains how antithetic MCMC can be more efficient
than i.i.d. sampling.
We also establish conditions that can guarantee that a method
is not efficiency-dominated by any other method.

Technical report, 29 May 2023 (revised 27 March 2024), 24 pages:
pdf.

Also available at
arXiv.org.

**Associated reference:**
The following is a companion paper that applies some of the theory developed:
Neal, R. M. (2024) ``Modifying Gibbs sampling to avoid self transitions'',
Technical Report, 84 pages:
abstract,
pdf.