Optimized Population Monte Carlo

Abstract

Adaptive importance sampling (AIS) methods are increasingly used for the approximation of distributions and related intractable integrals in the context of Bayesian inference. Population Monte Carlo (PMC) algorithms are a subclass of AIS methods, widely used due to their ease in the adaptation. In this paper, we propose a novel algorithm that exploits the benefits of the PMC framework and includes more efficient adaptive mechanisms, exploiting geometric information of the target distribution. In particular, the novel algorithm adapts the location and scale parameters of a set of importance densities (proposals). At each iteration, the location parameters are adapted by combining a versatile resampling strategy (i.e., using the information of previous weighted samples) with an advanced optimization-based scheme. Local second-order information of the target distribution is incorporated through a preconditioning matrix acting as a scaling metric onto a gradient direction. A damped Newton approach is adopted to ensure robustness of the scheme. The resulting metric is also used to update the scale parameters of the proposals. We discuss several key theoretical foundations for the proposed approach. Finally, we show the successful performance of the proposed method in three numerical examples, involving challenging distributions.