Establishing some order amongst exact approximations of MCMCs
Establishing some order amongst exact approximations of MCMCs
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DOI：
10.1214/15-aap1158
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发表时间：
2014-04
影响因子：
1.8
通讯作者：
C. Andrieu;M. Vihola
中科院分区：
文献类型：
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作者：
C. Andrieu;M. Vihola
马尔可夫链蒙特卡洛（MCMC）算法的精确近似是一般新兴的采样算法。精确近似值背后的主要思想之一是替换运行标准MCMC算法所需的棘手数量，例如大都市杂货店中的目标概率密度和估计量。也许令人惊讶的是，这样的近似值导致了强大的算法，从某种意义上说，它们可以保证具有正确的限制分布。在本文中，我们发现了一个通用框架，该框架可以比较或顺序的两种算法实现的性能度量。特别是，我们建立了关于平均接受概率，第一个自相关系数，渐近方差和右光谱差距的顺序。保证排序的关键概念是用于实现算法的估算器之间的凸顺序。我们认为，我们的凸顺序条件接近最佳状态，这是由反示例支持的，这表明较弱的方差顺序不够。凸顺序通过允许我们构建Martingale耦合来发挥核心作用，该耦合可以比较马尔可夫链的性能度量与不同的不变分布，这与现有结果相反。我们通过表明平均复制品以单调的方式提高性能，并保证该分层可以提高近似贝叶斯计算（ABC）MCMC方法的标准实施，从而详细介绍了结果的应用，并确保平均副本可以提高性能。
Exact approximations of Markov chain Monte Carlo (MCMC) algorithms are a general emerging class of sampling algorithms. One of the main ideas behind exact approximations consists of replacing intractable quantities required to run standard MCMC algorithms, such as the target probability density in a Metropolis-Hastings algorithm, with estimators. Perhaps surprisingly, such approximations lead to powerful algorithms which are exact in the sense that they are guaranteed to have correct limiting distributions. In this paper we discover a general framework which allows one to compare, or order, performance measures of two implementations of such algorithms. In particular, we establish an order with respect to the mean acceptance probability, the first autocorrelation coefficient, the asymptotic variance and the right spectral gap. The key notion to guarantee the ordering is that of the convex order between estimators used to implement the algorithms. We believe that our convex order condition is close to optimal, and this is supported by a counter-example which shows that a weaker variance order is not sufficient. The convex order plays a central role by allowing us to construct a martingale coupling which enables the comparison of performance measures of Markov chain with differing invariant distributions, contrary to existing results. We detail applications of our result by identifying extremal distributions within given classes of approximations, by showing that averaging replicas improves performance in a monotonic fashion and that stratification is guaranteed to improve performance for the standard implementation of the Approximate Bayesian Computation (ABC) MCMC method.