# Asymptotically exact data augmentation: models, properties and algorithms

## TL;DR

This paper introduces AXDA, a unified framework for data augmentation that maintains the original distribution asymptotically, offering statistical benefits and efficient inference algorithms, with theoretical analysis and practical illustrations.

## Contribution

It proposes a systematic approach to asymptotically exact data augmentation, unifying existing models and analyzing their statistical properties and computational efficiency.

## Key findings

- AXDA models preserve the original distribution asymptotically.
- Theoretical bounds on approximation quality are established.
- AXDA enables efficient inference algorithms with proven properties.

## Abstract

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 paper 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 paper 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 paper are available online.