Due to the potential benefits of parallelization, designing unbiased Monte Carlo estimators, primarily in the setting of randomized multilevel Monte Carlo, has recently become very popular in operations research and computational statistics. However, existing work primarily substantiates the benefits of unbiased estimators at an intuitive level or using empirical evaluations. The intuition being that unbiased estimators can be replicated in parallel enabling fast estimation in terms of wall-clock time. This intuition ignores that, typically, bias will be introduced due to impatience because most unbiased estimators necesitate random completion times. This paper provides a mathematical framework for comparing these methods under various metrics, such as completion time and overall computational cost. Under practical assumptions, our findings reveal that unbiased methods typically have superior completion times - the degree of superiority being quantifiable through the tail behavior of their running time distribution - but they may not automatically provide substantial savings in overall computational costs. We apply our findings to Markov Chain Monte Carlo and Multilevel Monte Carlo methods to identify the conditions and scenarios where unbiased methods have an advantage, thus assisting practitioners in making informed choices between unbiased and biased methods.

翻译：鉴于并行化的潜在优势，设计无偏蒙特卡洛估计器（主要应用于随机化多层蒙特卡洛框架）近年来在运筹学与计算统计学领域备受青睐。然而，现有研究主要基于直观推理或经验评估来论证无偏估计器的优势——其直觉逻辑在于无偏估计器可被并行复制，从而通过挂钟时间实现快速估计。这种直觉忽略了以下事实：由于多数无偏估计器必然产生随机完成时间，通常因"不耐烦"机制引入偏差。本文构建了数学分析框架，从完成时间与总体计算成本等多个维度比较两类方法。在实用假设条件下，我们的研究发现：无偏方法通常具有更优的完成时间（其优越程度可通过运行时分布的尾部行为量化），但未必能自动显著降低总体计算成本。我们将研究成果应用于马尔可夫链蒙特卡洛与多层蒙特卡洛方法，明确界定了无偏方法占据优势的条件与场景，为实践者在无偏与有偏方法间的明智抉择提供理论依据。