To improve the efficiency of Monte Carlo estimation, practitioners are turning to biased Markov chain Monte Carlo procedures that trade off asymptotic exactness for computational speed. The reasoning is sound: a reduction in variance due to more rapid sampling can outweigh the bias introduced. However, the inexactness creates new challenges for sampler and parameter selection, since standard measures of sample quality like effective sample size do not account for asymptotic bias. To address these challenges, we introduce a new computable quality measure based on Stein's method that bounds the discrepancy between sample and target expectations over a large class of test functions. We use our tool to compare exact, biased, and deterministic sample sequences and illustrate applications to hyperparameter selection, convergence rate assessment, and quantifying bias-variance tradeoffs in posterior inference.

为了提高 Monte Carlo 估计的效率，研究者们正转向有偏的马尔可夫链蒙特卡罗过程，通过权衡渐近精确度和计算速度来实现。本文引入一种基于 Stein's 方法的可计算质量度量来解决这些配合中不精确性带来的新挑战，并将其应用于超参数选择、收敛速率评估和后验推断中，比较精确、有偏和确定性样本序列，并量化样本和目标期望之间的最大偏差。

使用Stein方法测量样本质量