The data augmentation algorithm

The data augmentation (DA) algorithms are popular Markov chain Monte Carlo (MCMC) algorithms often used for sampling from intractable probability distributions. This review article comprehensively surveys DA MCMC algorithms, highlighting their theoretical foundations, methodological implementations, and diverse applications in frequentist and Bayesian statistics. The article discusses tools for studying the convergence properties of DA algorithms. Furthermore, it contains various strategies for accelerating the speed of convergence of the DA algorithms, different extensions of DA algorithms and outlines promising directions for future research. This paper aims to serve as a resource for researchers and practitioners seeking to leverage data augmentation techniques in MCMC algorithms by providing key insights and synthesizing recent developments.