A cluster-sample approach for Monte Carlo integration using multiple samplers

A computational problem in many fields is to estimate simultaneously multiple integrals and expectations, assuming that the data are generated by some Monte Carlo algorithm. Consider two scenarios in which draws are simulated from multiple distributions but the normalizing constants of those distributions may be known or unknown. For each scenario, existing estimators can be classified as using individual samples separately or using all the samples jointly. The latter pooled-sample estimators are statistically more efficient but computationally more costly to evaluate than the separate-sample estimators. We develop a cluster-sample approach to obtain computationally effective estimators, after draws are generated for each scenario. We divide all the samples into mutually exclusive clusters and combine samples from each cluster separately. Furthermore, we exploit a relationship between estimators based on samples from different clusters to achieve variance reduction. The resulting estimators, compared with the pooled-sample estimators, typically yield similar statistical efficiency but have reduced computational cost. We illustrate the value of the new approach by two examples for an Ising model and a censored Gaussian random field.