Optimal metropolis algorithms for product measures on the vertices of a hypercube

Optimal scaling problems for high dimensional Metropolis-Hastings algorithms can often be solved by means of diffusion approximation results. These solutions are particularly appealing since they can often be characterised in terms of a simple, observable property of the Markov chain sample path, namely the overall proportion of accepted iterations for the chain. For discrete state space problems, analogous scaling problems can be defined, though due to discrete effects, a simple characterisation of the asymptotically optimal solution is not available. This paper considers the simplest possible (and most discrete) example of such a problem, demonstrating that, at least for sufficiently 'smooth' distributions in high dimensional problems,the Metropolis algorithm behaves similarly to its counterpart on the continuous state space