Krzysztof Łatuszyński

Adaptive Gibbs samplers and related MCMC methods

We consider various versions of adaptive Gibbs and Metropolis- within-Gibbs samplers, which update their selection probabilities (and per- haps also their proposal distributions) on the y during a run, by learning as they go in an attempt to optimise the algorithm. We present a cautionary example of how even a simple-seeming adaptive Gibbs sampler may fail to converge. We then present various positive results guaranteeing convergence of adaptive Gibbs samplers under certain conditions. AMS 2000 subject classications: Primary 60J05, 65C05; secondary 62F15.

Variance bounding and geometric ergodicity of Markov chain Monte Carlo kernels for approximate Bayesian computation

Approximate Bayesian computation has emerged as a standard computational tool when dealing with intractable likelihood functions in Bayesian inference. We show that many common Markov chain Monte Carlo kernels used to facilitate inference in this setting can fail to be variance bounding and hence geometrically ergodic, which can have consequences for the reliability of estimates in practice. This phenomenon is typically independent of the choice of tolerance in the approximation. We prove that a recently introduced Markov kernel can inherit the properties of variance bounding and geometric ergodicity from its intractable Metropolis–Hastings counterpart, under reasonably weak conditions. We show that the computational cost of this alternative kernel is bounded whenever the prior is proper, and present indicative results for an example where spectral gaps and asymptotic variances can be computed, as well as an example involving inference for a partially and discretely observed, time-homogeneous, pure jump Markov process. We also supply two general theorems, one providing a simple sufficient condition for lack of variance bounding for reversible kernels and the other providing a positive result concerning inheritance of variance bounding and geometric ergodicity for mixtures of reversible kernels.

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 non-asymptotic. 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 �2 as(P; f)=n, where �2 as(P; f) is the CLT asymptotic variance. Next, we proceed to specific assumptions and give explicit computable bounds for geometrically and polynomially ergodic Markov chains. As a corollary we provide results on confidence estimation.

Simulating events of unknown probabilities via reverse time martingales

Let s∈(0,1) be uniquely determined but only its approximations can be obtained with a finite computational effort. Assume one aims to simulate an event of probability s. Such settings are often encountered in statistical simulations. We consider two specific examples. First, the exact simulation of non-linear diffusions ([3]). Second, the celebrated Bernoulli factory problem ([10, 13]) of generating an f(p)-coin given a sequence X1,X2,… of independent tosses of a p-coin (with known f and unknown p). We describe a general framework and provide algorithms where this kind of problems can be fitted and solved. The algorithms are straightforward to implement and thus allow for effective simulation of desired events of probability s. Our methodology links the simulation problem to existence and construction of unbiased estimators.

The containment condition and AdapFail algorithms

This short note investigates convergence of adaptive MCMC algorithms, i.e. algorithms which modify the Markov chain update probabilities on the fly. We focus on the Containment condition introduced in [RR07]. We show that if the Containment condition is not satisfied, then the algorithm will perform very poorly. Specifically, with positive probability, the adaptive algorithm will be asymptotically less efficient then any nonadaptive ergodic MCMC algorithm. We call such algorithms AdapFail, and conclude that they should not be used. AMS 2000 subject classifications: Primary 60J05, 65C05.

Stability of adversarial Markov Chains, with an application to adaptive MCMC algorithms

We consider whether ergodic Markov chains with bounded step size remain bounded in probability when their transitions are modied by an adversary on a bounded subset. We provide counterexamples to show that the answer is no in general, and prove theorems to show that the answer is yes under various additional assumptions. We then use our results to prove convergence of various adaptive Markov chain Monte Carlo algorithms.