Large deviation asymptotics and control variates for simulating large functions

Consider the normalized partial sums of a real-valued function $F$ of a Markov chain, $$\phi_n:=n^{-1}\sum_{k=0}^{n-1}F(\Phi(k)),\qquad n\ge1.$$ The chain $\{\Phi(k):k\ge0\}$ takes values in a general state space $\mathsf {X}$, with transition kernel $P$, and it is assumed that the Lyapunov drift condition holds: $PV\le V-W+b\mathbb{I}_C$ where $V:\mathsf {X}\to(0,\infty)$, $W:\mathsf {X}\to[1,\infty)$, the set $C$ is small and $W$ dominates $F$. Under these assumptions, the following conclusions are obtained: 1. It is known that this drift condition is equivalent to the existence of a unique invariant distribution $\pi$ satisfying $\pi(W)<\infty$, and the law of large numbers holds for any function $F$ dominated by $W$: $$\phi_n\to\phi:=\pi(F),\qquad{a.s.}, n\to\infty.$$ 2. The lower error probability defined by $\mathsf {P}\{\phi_n\le c\}$, for $c<\phi$, $n\ge1$, satisfies a large deviation limit theorem when the function $F$ satisfies a monotonicity condition. Under additional minor conditions an exact large deviations expansion is obtained. 3. If $W$ is near-monotone, then control-variates are constructed based on the Lyapunov function $V$, providing a pair of estimators that together satisfy nontrivial large asymptotics for the lower and upper error probabilities. In an application to simulation of queues it is shown that exact large deviation asymptotics are possible even when the estimator does not satisfy a central limit theorem.Comment: Published at http://dx.doi.org/10.1214/105051605000000737 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonasymptotic bounds on the estimation error of MCMC algorithms

We address the problem of upper bounding the mean square error of MCMC estimators. Our analysis is nonasymptotic. We first establish a general result valid for essentially all ergodic Markov chains encountered in Bayesian computation and a possibly unbounded target function $f$. The bound is sharp in the sense that the leading term is exactly $\sigma_{\mathrm {as}}^2(P,f)/n$, where $\sigma_{\mathrm{as}}^2(P,f)$ is the CLT asymptotic variance. Next, we proceed to specific additional assumptions and give explicit computable bounds for geometrically and polynomially ergodic Markov chains under quantitative drift conditions. As a corollary, we provide results on confidence estimation.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ442 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm). arXiv admin note: text overlap with arXiv:0907.491

Second-Order Coding Rates for Channels with State

We study the performance limits of state-dependent discrete memoryless channels with a discrete state available at both the encoder and the decoder. We establish the epsilon-capacity as well as necessary and sufficient conditions for the strong converse property for such channels when the sequence of channel states is not necessarily stationary, memoryless or ergodic. We then seek a finer characterization of these capacities in terms of second-order coding rates. The general results are supplemented by several examples including i.i.d. and Markov states and mixed channels