### Spectral bounds for certain two-factor non-reversible MCMC algorithms

Jeffrey Rosenthal, Peter Rosenthal
2015 Electronic Communications in Probability
We prove that the Markov operator corresponding to the two-variable, non-reversible Gibbs sampler has spectrum which is entirely real and non-negative, thus providing a first step towards the spectral analysis of MCMC algorithms in the non-reversible case. We also provide an extension to Metropolis-Hastings components, and connect the spectrum of an algorithm to the spectrum of its marginal chain. Introduction This paper is inspired by the earlier paper [23] , which discusses the importance of
more » ... eal, non-negative spectra for MCMC algorithms, and proves this property for several different reversible cases. In this paper, we extend that result to some common non-reversible MCMC algorithms, as we shall explain. Markov chain Monte Carlo (MCMC) algorithms, such as the Gibbs sampler [9, 8] and the Metropolis-Hastings algorithm [16, 10, 26] , are an extremely active area of modern research, with applications to numerous areas (see e.g. [3] and the references therein). Much of the mathematical analysis of these algorithms centers around their convergence rate; i.e., how long they need to be run before they produce accurate samples from the designated target probability distribution (cf. [20] ). Some of this analysis uses probabilistic techniques such as coupling and minorisation conditions (e.g. [21, 4] ). However, much of the analysis involves considering the spectrum of the associated Markov operator (see Section 2.2). In such cases, the Markov operator is nearly always assumed to be self-adjoint, corresponding to the Markov chain being reversible (see e.g. [13, 24, 6, 5, 12] ). The paradigm used is then roughly as follows: 1. Since the Markov operator is self-adjoint, its spectrum must be real (not complex), and can often be shown (or forced) to be non-negative, cf. [23]. 2. The corresponding spectral gap can then be bounded away from zero using various techniques (Cheeger's inequality, quadratic forms, etc.). 3. These spectral gap bounds then imply bounds on the operator's norm, which in turn lead to bounds on the Markov chain's convergence rate. However, if the Markov chain is not reversible, then much of this paradigm breaks down (though the spectral radius formula is still of some relevance to step 3 above; see Section 2.2 below), and the analysis becomes much more difficult (see e.g. [17] ). Some authors have attempted to get around this difficulty by replacing the non-reversible Markov chain by its "reversibilisation" [7], or by some other chain which provably has the same convergence properties [19] . However, there has been very little success at directly investigating the spectral properties of non-reversible Markov chains themselves, despite the fact that many commonly used MCMC algorithms (such as the systematic-scan Gibbs sampler) are not reversible and thus not amenable to the above paradigm.