Consistency of the Adaptive Multiple Importance Sampling

Among Monte Carlo techniques, the importance sampling requires fine tuning of a proposal distribution, which is now fluently resolved through iterative schemes. The Adaptive Multiple Importance Sampling (AMIS) of Cornuet et al. (2012) provides a significant improvement in stability and effective sample size due to the introduction of a recycling procedure. However, the consistency of the AMIS estimator remains largely open. In this work we prove the convergence of the AMIS, at a cost of a slight modification in the learning process. Contrary to Douc et al. (2007a), 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.