Consistency of adaptive importance sampling and recycling schemes

Abstract

Among Monte Carlo techniques, the importance sampling requires fine tuning of a proposal distribution, which is now fluently resolved through iterative schemes. Sequential adaptive algorithms have been proposed to calibrate the sampling distribution. Cornuet et al. [Scand. J. Stat. 39 (2012) 798–812] provides a significant improvement in stability and effective sample size by the introduction of a recycling procedure. However, the consistency of such algorithms have been rarely tackled because of their complexity. Moreover, the recycling strategy of the AMIS estimator adds another difficulty and its consistency remains largely open. In this work, we prove the convergence of sequential adaptive sampling, with finite Monte Carlo sample size at each iteration, and consistency of recycling procedures. Contrary to Douc et al. [Ann. Statist. 35 (2007) 420–448], results are obtained here in the asymptotic regime where the number of iterations is going to infinity while the number of drawings per iteration is a fixed, but growing sequence of integers. Hence, some of the results shed new light on adaptive population Monte Carlo algorithms in that last regime and give advices on how the sample sizes should be fixed.