关于某些自适应蒙特卡罗格式效率的警示
A cautionary tale on the efficiency of some adaptive Monte Carlo schemes
摘要：
在文献中，人们对基于随机转移核序列$\{P_n\}$的自适应马尔可夫链蒙特卡罗方法越来越感兴趣，其中允许核$P_n$具有不一定等于感兴趣的分布$π$（目标分布）的不变分布$π_n$。这些算法被设计为使得作为$n\to\infty$，$P_n$收敛到$P$，即具有正确不变分布$π$的核。通常，$P$是具有良好收敛特性的内核，但不能直接实现。然后期望该算法将继承$P$的良好收敛特性。[Ann. Statistist]的等能量采样器。34（2006）1581- 1619]是这种类型的自适应MCMC的示例。我们在本文中，这种类型的自适应MCMC的渐近方差总是至少与转移核$P$的马尔可夫链的渐近方差一样大。我们还通过模拟表明，差异可能是巨大的。
英文摘要：
There is a growing interest in the literature for adaptive Markov chain Monte Carlo methods based on sequences of random transition kernels $\{P_n\}$ where the kernel $P_n$ is allowed to have an invariant distribution $π_n$ not necessarily equal to the distribution of interest $π$ (target distribution). These algorithms are designed such that as $n\to\infty$, $P_n$ converges to $P$, a kernel that has the correct invariant distribution $π$. Typically, $P$ is a kernel with good convergence properties, but one that cannot be directly implemented. It is then expected that the algorithm will inherit the good convergence properties of $P$. The equi-energy sampler of [Ann. Statist. 34 (2006) 1581--1619] is an example of this type of adaptive MCMC. We show in this paper that the asymptotic variance of this type of adaptive MCMC is always at least as large as the asymptotic variance of the Markov chain with transition kernel $P$. We also show by simulation that the difference can be substantial.
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