Stochastic monotonicity and conditional Monte Carlo for likelihood ratios

Likelihood ratios are used in computer simulation to estimate expectations with respect to one law from simulation of another. This importance sampling technique can be implemented with either the likelihood ratio at the end of the simulated time horizon or with a sequence of likelihood ratios at intermediate times. Since a likelihood ratio process is a martingale, the intermediate values are conditional expectations of the final value and their use introduces no bias. We provide conditions under which using conditional expectations in this way brings guaranteed variance reduction. We use stochastic orderings to get positive dependence between a process and its likelihood ratio, from which variance reduction follows. Our analysis supports the following rough statement: for increasing functionals of associated processes with monotone likelihood ratio, conditioning helps. Examples are drawn from recursively defined processes, Markov chains in discrete and continuous time, and processes with Poisson input.