Monte Carlo algorithms represent the de facto standard for approximating complicated integrals involving multidimensional target distributions. In order to generate random realizations from the target distribution, Monte Carlo techniques use simpler proposal probability densities for drawing candidate samples. Performance of any such method is strictly related to the specification of the proposal distribution, such that unfortunate choices easily wreak havoc on the resulting estimators. In this work, we introduce a layered, that is a hierarchical, procedure for generating samples employed within a Monte Carlo scheme. This approach ensures that an appropriate equivalent proposal distribution is always obtained automatically (thus eliminating the risk of a catastrophic performance), although at the expense of a moderate increase in the complexity of the resulting algorithm. A hierarchical interpretation of two well-known methods, such as of the random walk Metropolis-Hastings (MH) and the Population Monte Carlo (PMC) techniques, is provided. Furthermore, we provide a general unified importance sampling (IS) framework where multiple proposal densities are employed, and several IS schemes are introduced applying the so-called deterministic mixture approach. Finally, given these schemes, we also propose a novel class of adaptive importance samplers using a population of proposals, where the adaptation is driven by independent parallel or interacting Markov Chain Monte Carlo (MCMC) chains. The resulting algorithms combine efficiently the benefits of both IS and MCMC methods.

本文介绍了一种使用分层（即分级）策略生成样本的蒙特卡罗方法，其中使用了多项提案密度和多种重要性采样方法，在保证性能的基础上，提出了一种新的自适应重要性采样算法。

分层自适应重要性采样