Asymptotically Exact Data Augmentation: Models, Properties, and Algorithms

Data augmentation, by the introduction of auxiliary variables, has become an ubiquitous technique to improve convergence properties, simplify the implementation or reduce the computational time of inference methods such as Markov chain Monte Carlo ones. Nonetheless, introducing appropriate auxiliary variables while preserving the initial target probability distribution and offering a computationally efficient inference cannot be conducted in a systematic way. To deal with such issues, this article studies a unified framework, coined asymptotically exact data augmentation (AXDA), which encompasses both well-established and more recent approximate augmented models. In a broader perspective, this article shows that AXDA models can benefit from interesting statistical properties and yield efficient inference algorithms. In non-asymptotic settings, the quality of the proposed approximation is assessed with several theoretical results. The latter are illustrated on standard statistical problems. Supplementary materials including computer code for this article are available online.

通过引入辅助变量的数据增强（data augmentation），已成为一项普适性技术，可用于改善马尔可夫链蒙特卡洛（Markov chain Monte Carlo）这类推断方法的收敛特性、简化实现流程或降低其计算耗时。然而，在保留初始目标概率分布的前提下引入恰当的辅助变量，并实现计算高效的推断，尚无法通过系统化方式完成。为解决此类问题，本文研究了一套统一框架，命名为渐近精确数据增强（asymptotically exact data augmentation，AXDA），该框架既涵盖了成熟的经典增强模型，也包含了近年提出的近似增强模型。从更广泛的视角来看，本文证明了渐近精确数据增强模型可具备优异的统计特性，并可衍生出高效的推断算法。在非渐近情形下，本文通过多项理论结果对所提近似方法的精度进行了评估，并将这些结果在标准统计问题中加以实例验证。本文的补充材料（含配套计算机代码）已在线公开。