Calibrated path sampling and stepwise bridge sampling

A computational problem in many fields is to evaluate multiple integrals and expectations simultaneously. Consider probability distributions with unnormalized density functions indexed by parameters on a 2-dimensional grid, and assume that samples are simulated from distributions on a subgrid. Examples of such unnormalized density functions include the observed-data likelihoods in the presence of missing data and the prior times the likelihood in Bayesian inference. There are various methods using a single sample only or multiple samples jointly to compute each integral. Path sampling seems a compromise, using samples along a 1-dimensional path to compute each integral. However, different choices of the path lead to different estimators, which should ideally be identical. We propose calibrated estimators by the method of control variates to exploit such constraints for variance reduction. We also propose biquadratic interpolation to approximate integrals with parameters outside the subgrid, consistently with the calibrated estimators on the subgrid. These methods can be extended to compute differences of expectations through an auxiliary identity for path sampling. Furthermore, we develop stepwise bridge-sampling methods in parallel but complementary to path sampling. In three simulation studies, the proposed methods lead to substantially reduced mean squared errors compared with existing methods.