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Consider a Markov chain defined on a finite state space, X, that leavesinvariant the uniform distribution on X, and whose transition probabilities areinteger multiples of 1/Q, for some integer Q. I show how a simulation of ntransitions of this chain starting at x_0 can be viewed as applying a randompermutation on the space XxU, where U={0,1,...,Q-1}, to the start state(x_0,u_0), with u_0 drawn uniformly from U. This result can be applied to anon-uniform distribution with probabilities that are integer multiples of 1/P,for some integer P, by representing it as the marginal distribution for X fromthe uniform distribution on a suitably-defined subset of XxY, whereY={0,1,...,P-1}. By letting Q, P, and the cardinality of X go to infinity, thisresult can be generalized to non-rational probabilities and to continuous statespaces, with permutations on a finite space replaced by volume-preservingone-to-one maps from a continuous space to itself. These constructions can beefficiently implemented for chains commonly used in Markov chain Monte Carlo(MCMC) simulations. I present two applications in this context - simulation ofK realizations of a chain from K initial states, but with transitions definedby a single stream of random numbers, as may be efficient with a vectorprocessor or multiple processors, and use of MCMC to improve an importancesampling distribution that already has substantial overlap with thedistribution of interest. I also discuss the implications of this "permutationMCMC" method regarding the role of randomness in MCMC simulation, and thepotential use of non-random and quasi-random numbers.

How to view an MCMC simulation as a permutation, with applications to parallel simulation and improved importance sampling