论文标题
数值MCMC的收敛速度和近似精度
Convergence Speed and Approximation Accuracy of Numerical MCMC
论文作者
论文摘要
在实施马尔可夫链蒙特卡洛（MCMC）算法时，有时是由数值错误引起的扰动。本文研究了MCMC的扰动如何影响收敛速度和蒙特卡洛估计准确性。我们的结果表明，当原始的马尔可夫链会收敛到平稳性足够快时，并且扰动的过渡内核是与原始过渡内核的良好近似值时，相应的扰动采样器也具有相似的收敛速度和高近似精度。我们讨论了两个不同的分析框架：终结性和光谱差距，两者都广泛用于文献中。我们的结果很容易扩展，以获得MCMC估计器的非反应误差界限。 我们还展示了如何将我们的收敛性和近似结果应用于特定采样算法的分析，包括随机行走大都市和大都市调整后的langevin算法，具有扰动的靶标密度，以及与扰动密度的平行回火蒙特卡洛。最后，我们提出了一些简单的数值示例，以验证我们的理论主张。
When implementing Markov Chain Monte Carlo (MCMC) algorithms, perturbation caused by numerical errors is sometimes inevitable. This paper studies how perturbation of MCMC affects the convergence speed and Monte Carlo estimation accuracy. Our results show that when the original Markov chain converges to stationarity fast enough and the perturbed transition kernel is a good approximation to the original transition kernel, the corresponding perturbed sampler has similar convergence speed and high approximation accuracy as well. We discuss two different analysis frameworks: ergodicity and spectral gap, both are widely used in the literature. Our results can be easily extended to obtain non-asymptotic error bounds for MCMC estimators. We also demonstrate how to apply our convergence and approximation results to the analysis of specific sampling algorithms, including Random walk Metropolis and Metropolis adjusted Langevin algorithm with perturbed target densities, and parallel tempering Monte Carlo with perturbed densities. Finally we present some simple numerical examples to verify our theoretical claims.
