In statistical analysis, Monte Carlo (MC) stands as a classical numerical integration method. When encountering challenging sample problem, Markov chain Monte Carlo (MCMC) is a commonly employed method. However, the MCMC estimator is biased after a fixed number of iterations. Unbiased MCMC, an advancement achieved through coupling techniques, addresses this bias issue in MCMC. However, its variance retains the traditional $O(N^{-1/2})$ convergence rate. Quasi-Monte Carlo (QMC), known for its high order of convergence, is an alternative of MC. By incorporating the idea of QMC into MCMC, Markov chain quasi-Monte Carlo (MCQMC) effectively reduces the variance of MCMC, especially in Gibbs samplers. This work presents a novel approach that integrates unbiased MCMC with MCQMC, called as an unbiased MCQMC method. This method renders unbiased estimators while improving the rate of convergence significantly. Numerical experiments demonstrate that the unbiased MCQMC method yields a substantial reduction in variance compared to unbiased MCMC in several Gibbs sampling problems. Particularly, unbiased MCQMC achieves convergence rates of approximately $O(N^{-1})$ in moderate dimensions.

翻译：在统计分析中，蒙特卡洛方法是一种经典的数值积分方法。当面临复杂采样问题时，马尔可夫链蒙特卡洛（MCMC）是常用方法。然而，固定迭代次数下MCMC估计量存在偏差。通过耦合技术实现的無偏MCMC解决了这一偏差问题，但其方差仍保持传统的$O(N^{-1/2})$收敛速度。拟蒙特卡洛（QMC）以其高阶收敛性著称，是MC的替代方法。通过将QMC思想融入MCMC，马尔可夫链拟蒙特卡洛（MCQMC）有效降低了MCMC方差，尤其在吉布斯采样器中。本文提出一种创新方法，将无偏MCMC与MCQMC相结合，称为无偏MCQMC方法。该方法在实现无偏估计量的同时显著提升收敛速度。数值实验表明，在多个吉布斯采样问题中，无偏MCQMC方法相比无偏MCMC能大幅降低方差。特别地，在中等维度下，无偏MCQMC可实现约$O(N^{-1})$的收敛速度。