February 2024 Approximating Bayes in the 21st Century
Gael M. Martin, David T. Frazier, Christian P. Robert
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Statist. Sci. 39(1): 20-45 (February 2024). DOI: 10.1214/22-STS875


The 21st century has seen an enormous growth in the development and use of approximate Bayesian methods. Such methods produce computational solutions to certain “intractable” statistical problems that challenge exact methods like Markov chain Monte Carlo: for instance, models with unavailable likelihoods, high-dimensional models and models featuring large data sets. These approximate methods are the subject of this review. The aim is to help new researchers in particular—and more generally those interested in adopting a Bayesian approach to empirical work—distinguish between different approximate techniques, understand the sense in which they are approximate, appreciate when and why particular methods are useful and see the ways in which they can can be combined.

Funding Statement

Martin and Frazier have been supported by Australian Research Council Discovery Grants DP170100729 and DP200101414, and the Australian Centre of Excellence in Mathematics and Statistics. Frazier has also been supported by Australian Research Council Discovery Early Career Researcher Award DE200101070. Robert has been partly supported by a senior chair (2016–2021) from l’Institut Universitaire de France and by a Prairie chair from the Agence Nationale de la Recherche (ANR-19-P3IA-0001).


Material that appeared in an earlier paper, entitled: ‘Computing Bayes: Bayesian Computation from 1763 to the 21st Century’ (Martin, Frazier and Robert, 2020), now appears (in amended form) in two separate papers: the current one, which provides a detailed review of 21st century approximate Bayesian methods, and a second one (Martin, Frazier and Robert, 2024) in which a historical overview of and timeline for all computational developments is presented. The authors would like to thank the Editor, an Associate Editor and two anonymous reviewers for very constructive and insightful comments on earlier drafts of the paper.


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Gael M. Martin. David T. Frazier. Christian P. Robert. "Approximating Bayes in the 21st Century." Statist. Sci. 39 (1) 20 - 45, February 2024. https://doi.org/10.1214/22-STS875


Published: February 2024
First available in Project Euclid: 18 February 2024

MathSciNet: MR4718525
Digital Object Identifier: 10.1214/22-STS875

Keywords: Approximate Bayesian Computation , Approximate Bayesian inference , Bayesian synthetic likelihood , Integrated nested Laplace approximation , intractable Bayesian problems , variational Bayes

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.39 • No. 1 • February 2024
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