We introduce a new property of Markov chains, called variance bounding. We prove that, for reversible chains at least, variance bounding is weaker than, but closely related to, geometric ergodicity. Furthermore, variance bounding is equivalent to the existence of usual central limit theorems for all $L^2$ functionals. Also, variance bounding (unlike geometric ergodicity) is preserved under the Peskun order. We close with some applications to Metropolis--Hastings algorithms.
Consideration is given to the three different analytical methods for the computation of upper bounds for the rate of convergence to the limiting regime of one specific class of (in)homogeneous continuous-time Markov chains. This class is particularly suited to describe evolutions of the total number of customers in (in)homogeneous $M/M/S$ queueing systems with possibly state-dependent arrival and service intensities, batch arrivals and services. One of the methods is based on the logarithmic norm of a linear operator function; the other two rely on Lyapunov functions and differential inequalit…
Bounding chains are a technique that offers three benefits to Markov chain practitioners: a theoretical bound on the mixing time of the chain under restricted conditions, experimental bounds on the mixing time of the chain that are provably accurate and construction of perfect sampling algorithms when used in conjunction with protocols such as coupling from the past. Perfect sampling algorithms generate variates exactly from the target distribution without the need to know the mixing time of a Markov chain at all. We present here the basic theory and use of bounding chains for several chains f…
The asymptotic variance is an important criterion to evaluate the performance of Markov chains, especially for the central limit theorems. We give the variational formulas for the asymptotic variance of discrete-time (non-reversible) Markov chains on general state space. The variational formulas provide many applications, extending the classical Peskun's comparison theorem to non-reversible Markov chains, and obtaining several comparison theorems between Markov chains with various perturbations.
A variational formula for the asymptotic variance of general Markov processes is obtained. As application, we get a upper bound of the mean exit time of reversible Markov processes, and some comparison theorems between the reversible and non-reversible diffusion processes.
We propose an exact technique to calculate lower bounds of spectral gaps of discrete time reversible Markov chains on finite state sets. Spectral gaps are a common tool for evaluating convergence rates of Markov chains. As an illustration, we successfully use this technique to evaluate the ``absorption time'' of the ``Backgammon model'', a paradigmatic model for glassy dynamics. We also discuss the application of this technique to the ``Contingency table problem'', a notoriously difficult problem from probability theory. The interest of this technique is that it connects spectral gaps, which a…
We obtain necessary and sufficient conditions for the regular variation of the variance of partial sums of functionals of discrete and continuous-time stationary Markov processes with normal transition operators. We also construct a class of Metropolis-Hastings algorithms which satisfy a central limit theorem and invariance principle when the variance is not linear in $n$.
We investigate an application of slowly varying functions (in sense of Karamata) in the theory of Markov branching process. We treat the critical case so that the infinitesimal generating function of the process has the infinite second moment, but it regularly varies with the remainder. We improve the Basic Lemma of the theory of critical Markov branching process and refine known limit results.
We derive sharper probabilistic concentration bounds for the Monte Carlo Empirical Rademacher Averages (MCERA), which are proved through recent results on the concentration of self-bounding functions. Our novel bounds are characterized by convergence rates that depend on data-dependent characteristic quantities of the set of functions under consideration, such as the empirical wimpy variance, an essential improvement w.r.t. standard bounds based on the methods of bounded differences. For this reason, our new results are applicable to yield sharper bounds to (Local) Rademacher Averages. We also…
Markov chain Monte Carlo(MCMC) is a popular approach to sample from high dimensional distributions, and the asymptotic variance is a commonly used criterion to evaluate the performance. While most popular MCMC algorithms are reversible, there is a growing literature on the development and analyses of nonreversible MCMC. Chen and Hwang(2013) showed that a reversible MCMC can be improved by adding an antisymmetric perturbation. They also raised a conjecture that it can not be improved if there is no cycle in the corresponding graph. In this paper, we present a rigorous proof of this conjecture. …
For a Markov transition kernel $P$ and a probability distribution $ μ$ on nonnegative integers, a time-sampled Markov chain evolves according to the transition kernel $P_μ = \sum_k μ(k)P^k.$ In this note we obtain CLT conditions for time-sampled Markov chains and derive a spectral formula for the asymptotic variance. Using these results we compare efficiency of Barker's and Metropolis algorithms in terms of asymptotic variance.
We introduce a simple iterative technique for bounding derivatives of solutions of Stein equations $Lf=h-\mathbb{E}h(Z)$, where $L$ is a linear differential operator and $Z$ is the limit random variable. Given bounds on just the solutions or certain lower order derivatives of the solution, the technique allows one to deduce bounds for derivatives of any order, in terms of supremum norms of derivatives of the test function $h$. This approach can be readily applied to many Stein equations from the literature. We consider a number of applications; in particular, we derive bounds for derivatives o…
In a previous paper we determined one dimensional distributions of a stationary field with linear regressions and quadratic conditional variances under a linear constraint on the coefficients of the quadratic expression. In this paper we show that for stationary Markov chains with linear regressions and quadratic conditional variances the coefficients of the quadratic expression are indeed tied by a linear constraint which can take only one of the two alternative forms.
We propose some backward-forward martingale decompositions for functions of reversible Markov chains. These decompositions are used to prove the functional CLT for reversible Markov chains with asymptotically linear variance of partial sums. We also provide a proof of the equivalence between asymptotic linearity of the variance and convergence of the integral of $1/(1-t)$ with respect to the associated spectral measure $ρ$. We also study the asymptotic behavior of linear processes having as innovations mean zero square integrable functions of stationary reversible Markov chains. We apply this …
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 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.
We prove deviation bounds for the random variable $\sum_{i=1}^{n} f_i(Y_i)$ in which $\{Y_i\}_{i=1}^{\infty}$ is a Markov chain with stationary distribution and state space $[N]$, and $f_i: [N] \rightarrow [-a_i, a_i]$. Our bound improves upon previously known bounds in that the dependence is on $\sqrt{a_1^2+\cdots+a_n^2}$ rather than $\max_{i}\{a_i\}\sqrt{n}.$ We also prove deviation bounds for certain types of sums of vector--valued random variables obtained from a Markov chain in a similar manner. One application includes bounding the expected value of the Schatten $\infty$-norm of a random…
We introduce the exit time finite state projection (ETFSP) scheme, a truncation-based method that yields approximations to the exit distribution and occupation measure associated with the time of exit from a domain (i.e., the time of first passage to the complement of the domain) of time-homogeneous continuous-time Markov chains. We prove that: (i) the computed approximations bound the measures from below; (ii) the total variation distances between the approximations and the measures decrease monotonically as states are added to the truncation; and (iii) the scheme converges, in the sense that…
In this paper we consider stationary Markov chains with trivial two-sided tail sigma field, and prove that additive functionals satisfy the central limit theorem provided the variance of partial sums divided by n is bounded.
We study the asymptotic behavior of kernel estimators of asymptotic variances (or long-run variances) for a class of adaptive Markov chains. The convergence is studied both in $L^p$ and almost surely. The results also apply to Markov chains and improve on the existing literature by imposing weaker conditions. We illustrate the results with applications to the $\operatorname {GARCH}(1,1)$ Markov model and to an adaptive MCMC algorithm for Bayesian logistic regression.
Using the renewal approach we prove Bernstein-like inequalities for additive functionals of geometrically ergodic Markov chains, thus obtaining counterparts of inequalities for sums of independent random variables. The coefficient in the sub-Gaussian part of our estimate is the asymptotic variance of the additive functional, i.e. the variance of the limiting Gaussian variable in the Central Limit Theorem for Markov chains. This refines earlier results by R. Adamczak and W. Bednorz, which were obtained under the additional assumption of strong aperiodicity of the chain.
The distribution of the "mixing time" or the "time to stationarity" in a discrete time irreducible Markov chain, starting in state i, can be defined as the number of trials to reach a state sampled from the stationary distribution of the Markov chain. Expressions for the probability generating function, and hence the probability distribution of the mixing time starting in state i are derived and special cases explored. This extends the results of the author regarding the expected time to mixing [J.J. Hunter, Mixing times with applications to perturbed Markov chains, Linear Algebra Appl. 417 (2…
In this paper we study the functional central limit theorem for stationary Markov chains with self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n; and establish that conditional convergence in distribution of partial sums implies functional CLT. The main tools are maximal inequalities that are further exploited to derive conditions for tightness and convergence to the Brownian motion.
In this paper it is shown that adaptive importance sampling algorithms converge at exponential rate for Markov chain expectation problems that admit a combination of a filtered estimator and a Markov zero-variance measure. It extends a chain of results---special purpose proofs were already known for several cases (Kollman et al, Baggerly et al, Desai). A recent paper (Awad et al) provides a complete description of the class of combinations of Markov process expectations of path functionals and filtered estimators that admit zero-variance importance measures that retain the Markov property. In …
In this paper we propose a novel variance reduction approach for additive functionals of Markov chains based on minimization of an estimate for the asymptotic variance of these functionals over suitable classes of control variates. A distinctive feature of the proposed approach is its ability to significantly reduce the overall finite sample variance. This feature is theoretically demonstrated by means of a deep non asymptotic analysis of a variance reduced functional as well as by a thorough simulation study. In particular we apply our method to various MCMC Bayesian estimation problems where…
We study the temporal dissipation of variance and relative entropy for ergodic Markov Chains in continuous time, and compute explicitly the corresponding dissipation rates. These are identified, as is well known, in the case of the variance in terms of an appropriate Hilbertian norm; and in the case of the relative entropy, in terms of a Dirichlet form which morphs into a version of the familiar Fisher information under conditions of detailed balance. Here we obtain trajectorial versions of these results, valid along almost every path of the random motion and most transparent in the backwards …
In this paper we study the additive functionals of Markov chains via conditioning with respect to both past and future of the chain. We shall point out new sufficient projective conditions, which assure that the variance of partial sums of n consecutive random variables of a stationary Markov chain is linear in n. The paper also addresses the central limit theorem problem and is listing several open questions.
We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, Randall, and Sinclair to generate random tilings of regions by lozenges. For an L X L region we bound the mixing time by O(L^4 log L), which improves on the previous bound of O(L^7), and we show the new bound to be essentially tight. In another application we resolve a few questions raised by Diaconis and Sal…
Optimal designs minimize the number of experimental runs (samples) needed to accurately estimate model parameters, resulting in algorithms that, for instance, efficiently minimize parameter estimate variance. Governed by knowledge of past observations, adaptive approaches adjust sampling constraints online as model parameter estimates are refined, continually maximizing expected information gained or variance reduced. We apply adaptive Bayesian inference to estimate transition rates of Markov chains, a common class of models for stochastic processes in nature. Unlike most previous studies, our…
The correspondence between the cross-entropy method and the zero-variance approximation to simulate a rare event problem in Markov chains is shown. This leads to a sufficient condition that the cross-entropy estimator is asymptotically optimal.
In order to sample from a given target distribution (often of Gibbs type), the Monte Carlo Markov chain method consists in constructing an ergodic Markov process whose invariant measure is the target distribution. By sampling the Markov process one can then compute, approximately, expectations of observables with respect to the target distribution. Often the Markov processes used in practice are time-reversible (i.e., they satisfy detailed balance), but our main goal here is to assess and quantify how the addition of a non-reversible part to the process can be used to improve the sampling prop…
We study convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms (Andrieu and Roberts [Ann. Statist. 37 (2009) 697-725]). We find that the asymptotic variance of the pseudo-marginal algorithm is always at least as large as that of the marginal algorithm. We show that if the marginal chain admits a (right) spectral gap and the weights (normalised estimates of the target density) are uniformly bounded, then the pseudo-marginal chain has a spectral gap. In many cases, a similar result holds for the absolute spectral gap, which is equivalent to geometric ergodicity. We consid…
We study control variate methods for Markov chain Monte Carlo (MCMC) in the setting of deterministic sweep sampling using $K\geq 2$ transition kernels. New variance reduction results are provided for MCMC averages based on sweeps over general transition kernels, leading to a particularly simple control variate estimator in the setting of deterministic sweep Gibbs sampling. Theoretical comparisons of our proposed control variate estimators with existing literature are made, and a simulation study is performed to examine the amount of variance reduction in some example cases. We also relate cont…
In this paper, we derive upper bounds on generalization errors for deep neural networks with Markov datasets. These bounds are developed based on Koltchinskii and Panchenko's approach for bounding the generalization error of combined classifiers with i.i.d. datasets. The development of new symmetrization inequalities in high-dimensional probability for Markov chains is a key element in our extension, where the spectral gap of the infinitesimal generator of the Markov chain plays a key parameter in these inequalities. We also propose a simple method to convert these bounds and other similar one…
We develop a general theory of Markov chains realizable as random walks on $\mathscr R$-trivial monoids. It provides explicit and simple formulas for the eigenvalues of the transition matrix, for multiplicities of the eigenvalues via Möbius inversion along a lattice, a condition for diagonalizability of the transition matrix and some techniques for bounding the mixing time. In addition, we discuss several examples, such as Toom-Tsetlin models, an exchange walk for finite Coxeter groups, as well as examples previously studied by the authors, such as nonabelian sandpile models and the promotion …
We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel and Toeplitz matrices generated by i.i.d. random variables $\{X_k\}$ of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables $\{X_{ij}\}_{j>i}$ of zero mean and unit variance, scaling the eigenvalues by $\sqrt{n}$ we prove the almost sure, weak convergence of the spectral measures to universal, nonrandom, symmetric distributions $γ_H$, $γ_M$ and $γ_T$ of unbounded support. The moments of $γ_H$ and $γ_T$ are the sum of volumes of solids related to …
