A popular framework for false discovery control is the random effects model in which the null hypotheses are assumed to be independent. This paper generalizes the random effects model to a conditional dependence model which allows dependence between null hypotheses. The dependence can be useful to characterize the spatial structure of the null hypotheses. Asymptotic properties of false discovery proportions and numbers of rejected hypotheses are explored and a large-sample distributional theory is obtained. 1. Introduction. Since

Given a Markov chain sampling scheme, does the standard empirical estimator make best use of the data? We show that this is not so and construct better estimators. We restrict attention to nearest neighbor random fields and to Gibbs samplers with deterministic sweep, but our approach applies to any sampler that uses reversible variable-at-a-time updating with deterministic sweep. The structure of the transition distribution of the sampler is exploited to construct further empirical estimators that are combined with the standard empirical estimator to reduce asymptotic variance. The extra computational cost is negligible. When the random field is spatially homogeneous, symmetrizations of our estimator lead to further variance reduction. The performance of the estimators is evaluated in a simulation study of the Ising model.

Suppose we want to calculate the expectation of a function f under a distribution on some space E. If E is of high dimension, or if is defined indirectly, it may be difficult to calculate the expectation f = E f = R (dx)f(x) analytically or even by numerical integration. The classical Monte Carlo method generates i.i.d. realizations X 0 ; : : : ; X n from , and approximates f by the empirical estimator E n f = 1 n n X i=1 f(X i ): If f is -integrable, the estimator is strongly consistent. If f is -square-