We design and analyze unbiased Markov chain Monte Carlo (MCMC) schemes based on couplings of blocked Gibbs samplers (BGSs), whose total computational costs scale linearly with the number of parameters and data points. Our methodology is designed for and applicable to high-dimensional BGS with conditionally independent blocks, which are often encountered in Bayesian modeling. We provide bounds on the expected number of iterations needed for coalescence for Gaussian targets, as well as on the tails of the coalescence times distribution. These imply that practical two-step coupling strategies achieve coalescence times that match the relaxation times of the original BGS scheme up to logarithmic factors. To illustrate the practical relevance of our methodology, we apply it to high-dimensional crossed random effect and probabilistic matrix factorization models, for which we develop a novel BGS scheme with improved convergence speed. Our methodology provides unbiased posterior estimates at linear cost (usually requiring only a few BGS iterations for problems with thousands of parameters), matching state-of-the-art procedures for both frequentist and Bayesian estimation of those models.

翻译：我们设计并分析了基于分块吉布斯采样器耦合的无偏马尔可夫链蒙特卡罗方案，其总计算成本随参数数量和数据点数量呈线性增长。该方法专为并适用于具有条件独立分块的高维分块吉布斯采样器，这类结构在贝叶斯建模中常见。针对高斯目标分布，我们给出了耦合收敛所需期望迭代次数的界，以及收敛时间分布的尾部界。这些结果表明，实用的两步耦合策略实现的收敛时间在至多对数因子内匹配原始分块吉布斯采样器的弛豫时间。为说明方法的实际价值，我们将其应用于高维交叉随机效应模型和概率矩阵分解模型，并为此开发了一种具有更快收敛速度的新型分块吉布斯采样器。我们的方法以线性成本提供无偏后验估计（对于含数千参数的问题通常仅需数次分块吉布斯采样迭代），其性能与针对这些模型的频率学派和贝叶斯估计的最先进方法相当。