ON THE CONVERGENCE OF ADAPTIVE SEQUENTIAL MONTE CARLO METHODS
ON THE CONVERGENCE OF ADAPTIVE SEQUENTIAL MONTE CARLO METHODS
复制标题
DOI：
10.1214/15-aap1113
复制
发表时间：
2016-04-01
影响因子：
1.8
通讯作者：
Thiery, Alexandre
中科院分区：
文献类型：
--
作者：
Beskos, Alexandros;Jasra, Ajay;Thiery, Alexandre
在顺序蒙特卡罗（SMC）方法的几种实现中，就算法效率而言，利用样本的历史信息来优化调整其后续传播是自然且重要的。在这篇文章中，我们提供了一个精心制定的渐近理论一类这样的自适应SMC方法。在假设下，这里开发的理论框架将覆盖几种常用的SMC算法[Chopin，Biometrika 89（2002）539-551; Jasra等人，《丑闻》杂志统计38（2011）1-22; Schafer和Chopin，Stat. Comput. 23（2013）163184]。关于这种自适应方法的理论基础，只有有限的结果：我们将通过为其中一些算法提供弱大数定律（WLLN）和中心极限定理（CLT）来弥合这一差距。后者似乎是文献中的第一个同类结果，并为许多真实的数据上下文中使用的算法提供了正式的理由[Jasra et al.（2011）; Schafer and Chopin（2013）]。我们建立，对于一般类的自适应SMC算法[肖邦（2002）]，自适应SMC方法的估计量的渐近方差是相同的“限制”SMC算法，使用理想的建议内核。我们的研究结果支持应用程序上的一个复杂的高维后验分布与Navier-Stokes模型，其中适应高维参数的建议内核是算法的效率至关重要。
In several implementations of Sequential Monte Carlo (SMC) methods it is natural and important, in terms of algorithmic efficiency, to exploit the information of the history of the samples to optimally tune their subsequent propagations. In this article we provide a carefully formulated asymptotic theory for a class of such adaptive SMC methods. The theoretical framework developed here will cover, under assumptions, several commonly used SMC algorithms [Chopin, Biometrika 89 (2002) 539-551; Jasra et al., Scand. J. Stat. 38 (2011) 1-22; Schafer and Chopin, Stat. Comput. 23 (2013) 163184]. There are only limited results about the theoretical underpinning of such adaptive methods: we will bridge this gap by providing a weak law of large numbers (WLLN) and a central limit theorem (CLT) for some of these algorithms. The latter seems to be the first result of its kind in the literature and provides a formal justification of algorithms used in many real data contexts [Jasra et al. (2011); Schafer and Chopin (2013)]. We establish that for a general class of adaptive SMC algorithms [Chopin (2002)], the asymptotic variance of the estimators from the adaptive SMC method is identical to a "limiting" SMC algorithm which uses ideal proposal kernels. Our results are supported by application on a complex high-dimensional posterior distribution associated with the Navier-Stokes model, where adapting high dimensional parameters of the proposal kernels is critical for the efficiency of the algorithm.