论文标题
使用耦合的马尔可夫链求解泊松方程
Solving the Poisson equation using coupled Markov chains
论文作者
论文摘要
本文展示了如何使用随机数的迭代来确切相遇的马尔可夫链，以生成泊松方程溶液的无偏估计量。通过这种联系，我们对马尔可夫链的固定分布进行了对预期的无偏估计量，并为其时刻的有限性提供了条件。我们进一步构建了马尔可夫链的渐近方差平均渐近方差的无偏估计量，并为估计器任何顺序的有限量提供条件。如果它们的第二刻是有限的，则此类估计量的独立副本的平均副本在蒙特卡洛速率下会收敛到渐近方差，与批次平均值和光谱方差估计器的已知速率相比，相比之下。通过数值实验说明了结果。
This article shows how coupled Markov chains that meet exactly after a random number of iterations can be used to generate unbiased estimators of the solutions of the Poisson equation. Through this connection, we re-derive known unbiased estimators of expectations with respect to the stationary distribution of a Markov chain and provide conditions for the finiteness of their moments. We further construct unbiased estimators of the asymptotic variance of Markov chain ergodic averages, and provide conditions for the finiteness of the estimators' moments of any order. If their second moment is finite, the average of independent copies of such estimators converges to the asymptotic variance at the Monte Carlo rate, comparing favorably to known rates for batch means and spectral variance estimators. The results are illustrated with numerical experiments.
