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August 2022 On the theoretical properties of the exchange algorithm
Guanyang Wang
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Bernoulli 28(3): 1935-1960 (August 2022). DOI: 10.3150/21-BEJ1405


The exchange algorithm is one of the most popular extensions of the Metropolis–Hastings algorithm to sample from doubly-intractable distributions. However, the theoretical exploration of the exchange algorithm is very limited. For example, natural questions like ‘Does exchange algorithm converge at a geometric rate?’ or ‘Does the exchange algorithm admit a Central Limit Theorem?’ have not been answered yet. In this paper, we study the theoretical properties of the exchange algorithm, in terms of asymptotic variance and convergence speed. We compare the exchange algorithm with the original Metropolis–Hastings algorithm and provide both necessary and sufficient conditions for the geometric ergodicity of the exchange algorithm. Moreover, we prove that our results can be applied to various practical applications such as location models, Gaussian models, Poisson models, and a large class of exponential families, which includes most of the practical applications of the exchange algorithm. A central limit theorem for the exchange algorithm is also established. Our results justify the theoretical usefulness of the exchange algorithm.


The author would like to thank Persi Diaconis, Julia Palacios, Wing H.Wong, and Daniel Rudolf for helpful discussions and comments. The author would like to thank the Editor, the Associate Editor and two referees for their constructive suggestions.


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Guanyang Wang. "On the theoretical properties of the exchange algorithm." Bernoulli 28 (3) 1935 - 1960, August 2022.


Received: 1 November 2020; Published: August 2022
First available in Project Euclid: 25 April 2022

Digital Object Identifier: 10.3150/21-BEJ1405

Keywords: convergence , geometrically ergodic , Markov chain Monte Carlo


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Vol.28 • No. 3 • August 2022
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