Asymptotic variance for random walk Metropolis chains in high dimensions: logarithmic growth via the Poisson equation
Asymptotic variance for random walk Metropolis chains in high dimensions: logarithmic growth via the Poisson equation
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DOI：
10.1017/apr.2019.40
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发表时间：
2017-07
影响因子：
1.2
通讯作者：
Aleksandar Mijatovi'c;Jure Vogrinc
中科院分区：
文献类型：
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作者：
Aleksandar Mijatovi'c;Jure Vogrinc
摘要 有两种加速马尔可夫链蒙特卡罗算法的方法：（a）构造更复杂的采样器，使用目标的梯度和高阶信息；（b）设计控制变量以减少渐近方差。虽然（a）的效率作为尺寸的函数已被广泛研究，但本文提供了将（b）的效率与尺寸联系起来的第一个结果。具体来说，我们使用[30]中缩放限制的泊松方程的解，为具有独立、同分布目标的 d 维随机游走 Metropolis 链构造一个控制变量。我们证明相应估计量的渐近方差在链的谱间隙上以 $\log(d)/d$ 的倍数为界。证明取决于大偏差理论、最优杨氏不等式和贝里-埃辛型界限。讨论了将结果扩展到非产品目标。
Abstract There are two ways of speeding up Markov chain Monte Carlo algorithms: (a) construct more complex samplers that use gradient and higher-order information about the target and (b) design a control variate to reduce the asymptotic variance. While the efficiency of (a) as a function of dimension has been studied extensively, this paper provides the first results linking the efficiency of (b) with dimension. Specifically, we construct a control variate for a d-dimensional random walk Metropolis chain with an independent, identically distributed target using the solution of the Poisson equation for the scaling limit in [30]. We prove that the asymptotic variance of the corresponding estimator is bounded above by a multiple of $\log(d)/d$ over the spectral gap of the chain. The proof hinges on large deviations theory, optimal Young’s inequality and Berry–Esseen-type bounds. Extensions of the result to non-product targets are discussed.