MCMC simulation as a random permutation

I’ve just finished a new paper. Continuing my recent use of unwieldy titles, I call it “How to view an MCMC simulation as a permutation, with applications to parallel simulation and improved importance sampling”.

The paper may look a bit technical in places, but the basic idea is fairly simple. I show that, after extending the state space a bit, it’s possible to view an MCMC simulation (done for some number of iterations) as a randomly selected map from an initial state to a final state that is either a permutation, if the extended state space is finite, or more generally a one-to-one map that preserves volume.

Why is this interesting? I think it’s a useful mathematical fact — sort of the opposite of how one can “couple” MCMC simulations in a way that promotes coalescence of states. It may turn out to be applicable in many contexts. I present two of these in the paper.

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