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Optimal Metropolis algorithms for product measures on the vertices of a hypercube. (English) Zbl 0904.60021
Summary: 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 characterized 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 characterization 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.
MSC:
60F05Central limit and other weak theorems
65C99Probabilistic methods, simulation and stochastic differential equations (numerical analysis)