On the geometric ergodicity of Metropolis-Hastings algorithms

Abstract

Under a compactness assumption, we show that a φ-irreducible and aperiodic Metropolis-Hastings chain is geometrically ergodic if and only if its rejection probability is bounded away from unity. In the particular case of the independence Metropolis-Hastings algorithm, we obtain that the whole spectrum of the induced operator is contained in (and in many cases equal to) the essential range of the rejection probability of the chain as conjectured by [Liu, J.S., 1996, Metropolized independent sampling with comparaisons to rejection sampling and importance sampling. Statistics and Computing, 6, 113–119.].

Keywords:

• Geometric ergodicity
• Markov chain operators
• Metropolis–Hastings algorithm

Acknowledgements

We are grateful to the editor and an anonymous referee for their very helpful comments and for improving the presentation of this paper. This work has been partly supported by NSERC Canada and ISM Montreal.