Non-reversible Markov Chains Research Articles

Optimizing the asymptotic convergence rate of the Diaconis–Holmes–Neal sampler

We consider the problem of optimizing the asymptotic convergence rate of a parameter-dependent nonreversible Markov chain. We begin with a single-parameter case studied by Diaconis, Holmes and Neal and then introduce multiple parameters. We use nonsmooth analysis to investigate whether the presence of multiple parameters allows a faster asymptotic convergence rate, and argue that for a specific parameterization, it does not, at least locally.

Laplacians and the Cheeger Inequality for Directed Graphs

We consider Laplacians for directed graphs and examine their eigenvalues. We introduce a notion of a circulation in a directed graph and its connection with the Rayleigh quotient. We then define a Cheeger constant and establish the Cheeger inequality for directed graphs. These relations can be used to deal with various problems that often arise in the study of non-reversible Markov chains including bounding the rate of convergence and deriving comparison theorems.

Card shuffling and the decomposition of tensor products

Let H be a subgroup of a .nite group G. We use Markov chains to quantify how large r should be so that the decomposition of the r tensor power of the representation of G on cosets on H behaves (after renormalization) like the regular representation of G. For the case where G is a symmetric group and H a parabolic subgroup, we .nd that this question is precisely equivalent to the question of how large r should be so that r iterations of a shu.ing method randomise the Robinson-Schensted-Knuth shape of a permutation. This equivalence is remarkable, if only because the representation theory problem is related to a reversible Markov chain on the set of representations of the symmetric group, whereas the card shu.ing problem is related to a nonreversible Markov chain on the symmetric group. The equivalence is also useful, and results on card shu.ing can be applied to yield sharp results about the decomposition of tensor powers.

A local limit theorem for a family of non-reversible Markov chains

By proving a local limit theorem for higher-order transitions, we determine the time required for necklace chains to be close to stationarity. Because necklace chains, built by arranging identical smaller Markov chains around a directed cycle, are not reversible, have little symmetry, do not have uniform stationary distributions, and can be nearly periodic, prior general bounds on rates of convergence of Markov chains either do not apply or give poor bounds. Necklace chains can serve as test cases for future techniques for bounding rates of convergence.

Analysis of a nonreversible Markov chain sampler

We analyze the convergence to stationarity of a simple nonreversible Markov chain that serves as a model for several nonreversible Markov chain sampling methods that are used in practice. Our theoretical and numerical results show that nonreversibility can indeed lead to improvements over the diffusive behavior of simple Markov chain sampling schemes. The analysis uses both probabilistic techniques and an explicit diagonalization.

The Collapsed Gibbs Sampler in Bayesian Computations with Applications to a Gene Regulation Problem

Abstract This article describes a method of “grouping” and “collapsing” in using the Gibbs sampler and proves from an operator theory viewpoint that the method is in general beneficial. The norms of the forward operators associated with the corresponding nonreversible Markov chains are used to discriminate among different simulation schemes. When applied to Bayesian missing data problems, the idea of collapsing suggests skipping the steps of sampling parameter(s) values in standard data augmentation. By doing this, we obtain a predictive update version of the Gibbs sampler. A procedure of calculating the posterior odds ratio via the collapsed Gibbs sampler when incomplete observations are involved is presented. As an illustration of possible applications, three examples, along with a Bayesian treatment for identifying common protein binding sites in unaligned DNA sequences, are provided.

Insensitive bounds for the stationary distribution of non-reversible Markov chains

A general method is developed to compute easy bounds of the weighted stationary probabilities for networks of queues which do not satisfy the standard product form. The bounds are obtained by constructing approximating reversible Markov chains. Thus, the bounds are insensitive with respect to service-time distributions. A special representation, called the tree-form solution, of the stationary distribution is used to derive the bounds. The results are applied to an overflow model.

Insensitive bounds for the stationary distribution of non-reversible Markov chains

A general method is developed to compute easy bounds of the weighted stationary probabilities for networks of queues which do not satisfy the standard product form. The bounds are obtained by constructing approximating reversible Markov chains. Thus, the bounds are insensitive with respect to service-time distributions. A special representation, called the tree-form solution, of the stationary distribution is used to derive the bounds. The results are applied to an overflow model.