Recent decades have seen enormous improvements in computational inference for statistical models; there have been competitive continual enhancements in a wide range of computational tools. In Bayesian inference, first and foremost, MCMC techniques have continued to evolve, moving from random walk proposals to Langevin drift, to Hamiltonian Monte Carlo, and so on, with both theoretical and algorithmic innovations opening new opportunities to practitioners. However, this impressive evolution in capacity is confronted by an even steeper increase in the complexity of the datasets to be addressed. The difficulties of modelling and then handling ever more complex datasets most likely call for a new type of tool for computational inference that dramatically reduces the dimension and size of the raw data while capturing its essential aspects. Approximate models and algorithms may thus be at the core of the next computational revolution.

We consider various versions of adaptive Gibbs and Metropolis-within-Gibbs samplers, which update their selection probabilities (and perhaps also their proposal distributions) on the fly during a run by learning as they go in an attempt to optimize 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.Comment: Published in at http://dx.doi.org/10.1214/11-AAP806 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: substantial text overlap with arXiv:1001.279

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

Approximate Bayesian computation has emerged as a standard computational tool when dealing with the increasingly common scenario of completely 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 then prove that a recently introduced Markov kernel in this setting can inherit variance bounding and geometric ergodicity from its intractable Metropolis-Hastings counterpart, under reasonably weak and manageable conditions. We show that the computational cost of this alternative kernel is bounded whenever the prior is proper, and present indicative results on an example where spectral gaps and asymptotic variances can be computed, as well as an example involving inference for a partially and discretely observed, timehomogeneous, pure jump Markov process. We also supply two general theorems, one of which provides a simple sufficient condition for lack of variance bounding for reversible kernels and the other provides a positive result concerning inheritance of variance bounding and geometric ergodicity for mixtures of reversible kernels.