Von mises type statistics for single site updated local interaction randomfields

Random field models in image analysis and spatial statistics usually have l ocal interactions. They can be simulated by Markov chains which update a si ngle site at a time. The updating rules typically condition on only a few n eighboring sites. If we want to approximate the expectation of a bounded fu nction, can we make better use of the simulations than through the empirica l estimator? We describe symmetrizations of the empirical estimator which a re computationally feasible and can lead to considerable variance reduction . The method is reminiscent of the idea behind generalized von Mises statis tics. To simplify the exposition, we consider mainly nearest neighbor rando m fields and the Gibbs sampler.