Splitting Markov fields and combining empirical estimators

We have shown elsewhere that the empirical estimator for the expectation of a local function on a Markov field over a lattice is ecient if and only if the function is a sum of functions each of which depends only on the values of the field on a clique of sites. For countable state space, the estimation of such expectations reduces to the estimation of probabilities of configurations over finite subsets of the lattice. The corresponding empirical estimator is ecient if and only if the set is a clique. If the set is not a clique, we can construct better estimators. They are rational functions of empirical estimators for configurations over subsets of the given set. The construction is based on a factorization of probabilities of configurations which makes use of the splitting property of Markov fields. AMS 1991 subject classifications. 62G05, 62M40.