Markov chain Monte Carlo (MCMC) is a commonly used method for approximating expectations with respect to probability distributions. Uncertainty assessment for MCMC estimators is essential in practical applications. Moreover, for multivariate functions of a Markov chain, it is important to estimate not only the auto-correlation for each component but also to estimate cross-correlations, in order to better assess sample quality, improve estimates of effective sample size, and use more effective stopping rules. Berg and Song [2022] introduced the moment least squares (momentLS) estimator, a shape-constrained estimator for the autocovariance sequence from a reversible Markov chain, for univariate functions of the Markov chain. Based on this sequence estimator, they proposed an estimator of the asymptotic variance of the sample mean from MCMC samples. In this study, we propose novel autocovariance sequence and asymptotic variance estimators for Markov chain functions with multiple components, based on the univariate momentLS estimators from Berg and Song [2022]. We demonstrate strong consistency of the proposed auto(cross)-covariance sequence and asymptotic variance matrix estimators. We conduct empirical comparisons of our method with other state-of-the-art approaches on simulated and real-data examples, using popular samplers including the random-walk Metropolis sampler and the No-U-Turn sampler from STAN.

翻译：马尔可夫链蒙特卡洛（MCMC）是一种常用于近似概率分布期望的方法。在实际应用中，对MCMC估计量进行不确定性评估至关重要。此外，对于马尔可夫链的多元函数，不仅需要估计各分量的自相关性，还需估计交叉相关性，以更好地评估样本质量、改进有效样本量的估计，并采用更有效的停止规则。Berg与Song [2022] 针对马尔可夫链的单变量函数，提出了矩最小二乘（momentLS）估计器——一种用于可逆马尔可夫链自协方差序列的形状约束估计器。基于该序列估计器，他们提出了MCMC样本均值渐近方差的估计量。在本研究中，我们基于Berg与Song [2022] 的单变量momentLS估计器，针对具有多分量的马尔可夫链函数，提出了新颖的自协方差序列与渐近方差估计器。我们证明了所提出的自（交叉）协方差序列估计器与渐近方差矩阵估计器的强相合性。我们通过模拟和真实数据案例，将所提方法与其他前沿方法进行了实证比较，所使用的采样器包括随机游走Metropolis采样器及STAN中的No-U-Turn采样器。