On the performance of nonlinear importance samplers and population Monte Carlo schemes

摘要

We investigate the estimation of normalisation constants of probability distributions using nonlinear importance sampling (IS). This is a problem that involves the solution of complicated multidimensional integrals and, in general, does not admit a closed-form solution or approximation. It is especially relevant for Bayesian model assessment problems, where the normalisation constant of the posterior distribution of the model parameters yields the model likelihood or model evidence. In this paper we prove that the clipped weights used in nonlinear IS yield a reduction of the variance of the resulting estimators of the normalisation constant (compared to standard IS). Then we derive a rule, based on the Pearson divergence between probability distributions, for the update of the proposal densities in population Monte Carlo algorithms (a class of adaptive IS schemes). The rule is designed to decrease the upper bound on the estimation error of the normalisation constant each time we update the proposal density.