Single-chain Markov chain Monte Carlo simulates realizations from a Markov chain to estimate expectations with the empirical average. The single-chain simulation is generally of considerable length and restricts many advantages of modern parallel computation. This paper constructs a novel many-short-chains Monte Carlo (MSC) estimator by averaging over multiple independent sums from Markov chains of a guaranteed short length. The computational advantage is the independent Markov chain simulations can be fast and may be run in parallel. The MSC estimator requires an importance sampling proposal and a drift condition on the Markov chain without requiring convergence analysis on the Markov chain. A non-asymptotic error analysis is developed for the MSC estimator under both geometric and multiplicative drift conditions. Empirical performance is illustrated on an autoregressive process and the P\'olya-Gamma Gibbs sampler for Bayesian logistic regression to predict cardiovascular disease.

翻译：单链马尔可夫链蒙特卡洛方法通过模拟一条马尔可夫链的样本轨迹，利用经验均值估计期望值。然而，单链模拟通常需要极长的链长，从而限制了现代并行计算的诸多优势。本文构建了一种新颖的多条短链蒙特卡洛（MSC）估计器，通过对多条保证短长度的马尔可夫链的独立求和结果进行平均来获得估计。其计算优势在于，各条马尔可夫链的模拟可独立快速执行，并支持并行运行。该MSC估计器依赖于重要性采样提议分布以及马尔可夫链的漂移条件，无需对马尔可夫链进行收敛性分析。在几何漂移条件和乘法漂移条件下，本文为MSC估计器建立了非渐近误差分析。通过自回归过程以及用于预测心血管疾病的贝叶斯逻辑回归的Pólya-Gamma吉布斯采样器，验证了该方法的经验性能。