Abstract
The classical Metropolis-Hastings (MH) algorithm can be extended to generate non-reversible Markov chains. This is achieved by means of a modification of the acceptance probability, using the notion of vorticity matrix. The resulting Markov chain is non-reversible. Results from the literature on asymptotic variance, large deviations theory and mixing time are mentioned, and in the case of a large deviations result, adapted, to explain how non-reversible Markov chains have favorable properties in these respects. We provide an application of NRMH in a continuous setting by developing the necessary theory and applying, as first examples, the theory to Gaussian distributions in three and nine dimensions. The empirical autocorrelation and estimated asymptotic variance for NRMH applied to these examples show significant improvement compared to MH with identical stepsize.
Highlights
The Metropolis-Hastings (MH) algorithm Metropolis et al (1953), Hastings (1970) is a Markov chain Monte-Carlo (MCMC) method of profound importance to many fields of mathematics such as Bayesian inference and statisti-cal mechanics Diaconis and Saloff-Coste (1998), Diaconis (2008), Levin et al (2009)
The chains generated by the classical Metropolis-Hastings algorithm are reversible, or, in other words, satisfy detailed balance; this reversibility is instrumental in showing that the resulting chains have the right invariant probability distribution
In this paper MH is extended to ‘non-reversible Metropolis-Hastings’ (NRMH) which allows for non-reversible transitions
Summary
The Metropolis-Hastings (MH) algorithm Metropolis et al (1953), Hastings (1970) is a Markov chain Monte-Carlo (MCMC) method of profound importance to many fields of mathematics such as Bayesian inference and statisti-. In this paper we consider the second type of creating non-reversibility, i.e. without augmenting the state space In discrete spaces this can, in principle, be achieved by changing transition probabilities (see Remark 2.2). Stat Comput (2016) 26:1213–1228 analogue in continuous spaces (crudely speaking, because all transition probabilities to specific states are zero) To remedy these issues, in this paper MH is extended to ‘non-reversible Metropolis-Hastings’ (NRMH) which allows for non-reversible transitions. The theory is first developed for discrete state spaces It is shown how the acceptance probability of MH, can be adjusted so that the resulting chain in NRMH has a specified ‘vorticity’, and will be non-reversible. Theoretical advantages of finite state space non-reversible chains in terms of improved asymptotic variance and large deviations estimates are briefly mentioned in Sect.
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