论文标题
估计来自多个马尔可夫链的蒙特卡洛差异
Estimating Monte Carlo variance from multiple Markov chains
论文作者
论文摘要
现代计算进步已经实现了马尔可夫链蒙特卡洛（MCMC）的简单平行实现。但是，几乎所有在估计蒙特卡洛平均值的方差（包括有效批次平均值（BM）估计器）方面的工作都集中在单链MCMC运行上。我们证明，简单地平均多个链中的协方差矩阵估计器可以在小蒙特卡洛样本量中产生关键的低估，尤其是对于慢速混合马尔可夫链。我们扩展了\ cite {arg：and：2006}的工作，并提出了一个多变量重复的批处理均值（RBM）估计器，该估计器利用了来自平行链中的信息，从而纠正了低估。在该过程混合速率的弱条件下，RBM非常一致，并且与BM估计量相似的大样本偏差和方差相似。我们还表明，在MCMC中存在正相关的情况下，RBM估计量中的（负）偏差小于平均BM估计量，我们表现出了RBM的卓越理论特性。因此，在小型运行中，RBM估计量可以非常出色，这可以通过各种示例来证明。
Modern computational advances have enabled easy parallel implementations of Markov chain Monte Carlo (MCMC). However, almost all work in estimating the variance of Monte Carlo averages, including the efficient batch means (BM) estimator, focuses on a single-chain MCMC run. We demonstrate that simply averaging covariance matrix estimators from multiple chains can yield critical underestimates in small Monte Carlo sample sizes, especially for slow-mixing Markov chains. We extend the work of \cite{arg:and:2006} and propose a multivariate replicated batch means (RBM) estimator that utilizes information from parallel chains, thereby correcting for the underestimation. Under weak conditions on the mixing rate of the process, RBM is strongly consistent and exhibits similar large-sample bias and variance to the BM estimator. We also exhibit superior theoretical properties of RBM by showing that the (negative) bias in the RBM estimator is less than the average BM estimator in the presence of positive correlation in MCMC. Consequently, in small runs, the RBM estimator can be dramatically superior and this is demonstrated through a variety of examples.
