Variational method for estimating the rate of convergence of Markov-chain Monte Carlo algorithms.
Variational method for estimating the rate of convergence of Markov-chain Monte Carlo algorithms.
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用于估计马尔可夫链蒙特卡罗算法收敛速度的变分方法。
DOI：
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发表时间：
2006
期刊：
影响因子：
--
通讯作者：
J. Sethna
中科院分区：
文献类型：
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作者：
F. Casey;J. Waterfall;R. Gutenkunst;C. Myers;J. Sethna
我们使用变分方法来确定作为目标密度和建议密度的函数的马尔可夫链蒙特卡罗(MCMC)算法的收敛速度的一个定量下界。这个界依赖于用变分原理逼近MCMC算子谱中的第二大本征值，并且该方法适用于具有连续状态空间的问题。我们将该方法应用于具有高斯和四次目标密度的一维实例，并将随机游走的Metropolis-Hastings算法与将梯度信息结合到试移中的“智能”变体的性能进行了对比，这是Metropolis调整的朗之万算法的推广。我们发现，变分方法与数值模拟非常接近。我们还看到，智能的MCMC算法通常不能在目标密度的尾部几何收敛，除非在我们检查的最简单的情况下，而且即使在这样的情况下，也必须谨慎地选择所提议的移动的确定性和随机部分的适当比例。这再次让人质疑智能MCMC在更复杂问题上的效用。最后，我们将同样的方法应用于有重要抽样和无重要抽样的多维高斯问题的收敛速度的逼近。在那里，我们论证了对目标密度进行重要抽样的必要性，目标密度依赖于大范围尺度的变量。
We demonstrate the use of a variational method to determine a quantitative lower bound on the rate of convergence of Markov chain Monte Carlo (MCMC) algorithms as a function of the target density and proposal density. The bound relies on approximating the second largest eigenvalue in the spectrum of the MCMC operator using a variational principle and the approach is applicable to problems with continuous state spaces. We apply the method to one dimensional examples with Gaussian and quartic target densities, and we contrast the performance of the random walk Metropolis-Hastings algorithm with a "smart" variant that incorporates gradient information into the trial moves, a generalization of the Metropolis adjusted Langevin algorithm. We find that the variational method agrees quite closely with numerical simulations. We also see that the smart MCMC algorithm often fails to converge geometrically in the tails of the target density except in the simplest case we examine, and even then care must be taken to choose the appropriate scaling of the deterministic and random parts of the proposed moves. Again, this calls into question the utility of smart MCMC in more complex problems. Finally, we apply the same method to approximate the rate of convergence in multidimensional Gaussian problems with and without importance sampling. There we demonstrate the necessity of importance sampling for target densities which depend on variables with a wide range of scales.
坚固的 Metropolis 采样，同时更新两个动态变量。
DOI：
10.1103/physreve.72.016712
发表时间：
2005
期刊：
Physical review. E, Statistical, nonlinear, and soft matter physics
影响因子：
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作者：
Berg,BerndA;Zhou,Huan-Xiang
通讯作者：
Zhou,Huan-Xiang
草率模型普遍性类和范德蒙德矩阵。
影响因子：
8.6
作者：
Waterfall,JoshuaJ;Casey,FergalP;Gutenkunst,RyanN;Brown,KevinS;Myers,ChristopherR;Brouwer,PietW;Elser,Veit;Sethna,JamesP
通讯作者：
Sethna,JamesP