We introduce a new class of sequential Monte Carlo methods called Nested Sampling via Sequential Monte Carlo (NS-SMC), which reframes the Nested Sampling method of Skilling (2006) in terms of sequential Monte Carlo techniques. This new framework allows convergence results to be obtained in the setting when Markov chain Monte Carlo (MCMC) is used to produce new samples. An additional benefit is that marginal likelihood estimates are unbiased. In contrast to NS, the analysis of NS-SMC does not require the (unrealistic) assumption that the simulated samples be independent. As the original NS algorithm is a special case of NS-SMC, this provides insights as to why NS seems to produce accurate estimates despite a typical violation of its assumptions. For applications of NS-SMC, we give advice on tuning MCMC kernels in an automated manner via a preliminary pilot run, and present a new method for appropriately choosing the number of MCMC repeats at each iteration. Finally, a numerical study is conducted where the performance of NS-SMC and temperature-annealed SMC is compared on several challenging and realistic problems. MATLAB code for our experiments is made available online at https://github.com/LeahPrice/SMC-NS.

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Sequential Monte Carlo (SMC) methods for sampling from the posterior of static Bayesian models are flexible, parallelisable and capable of handling complex targets. However, it is common practice to adopt a Markov chain Monte Carlo (MCMC) kernel with a multivariate normal random walk (RW) proposal in the move step, which can be both inefficient and detrimental for exploring challenging posterior distributions. We develop new SMC methods with independent proposals which allow recycling of all candidates generated in the SMC process and are embarrassingly parallelisable. A novel evidence estimator that is easily computed from the output of our independent SMC is proposed. Our independent proposals are constructed via flexible copula-type models calibrated with the population of SMC particles. We demonstrate through several examples that more precise estimates of posterior expectations and the marginal likelihood can be obtained using fewer likelihood evaluations than the more standard RW approach.

Zero-variance control variates (ZV-CV) are a post-processing method to reduce the variance of Monte Carlo estimators of expectations using the derivatives of the log target. Once the derivatives are available, the only additional computational effort is solving a linear regression problem. Significant variance reductions have been achieved with this method in low dimensional examples, but the number of covariates in the regression rapidly increases with the dimension of the target. We propose to exploit penalised regression to make the method more flexible and feasible, particularly in higher dimensions. Connections between this penalised ZV-CV approach and control functionals are made, providing additional motivation for our approach. Another type of regularisation based on using subsets of derivatives, or a priori regularisation as we refer to it in this paper, is also proposed to reduce computational and storage requirements. Methods for applying ZV-CV and regularised ZV-CV to sequential Monte Carlo (SMC) are described and a new estimator for the normalising constant of the posterior is developed to aid Bayesian model choice. Several examples showing the utility and limitations of regularised ZV-CV for Bayesian inference are given. The methods proposed in this paper are accessible through the R package ZVCV available at https://github.com/LeahPrice/ZVCV.

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This paper focuses on the numerical computation of posterior expected quantities of interest, where existing approaches based on ergodic averages are gated by the asymptotic variance of the integrand. To address this challenge, a novel variance reduction technique is proposed, based on Sard's approach to numerical integration and the control functional method. The use of Sard's approach ensures that our control functionals are exact on all polynomials up to a fixed degree in the Bernstein-von-Mises limit, so that the reduced variance estimator approximates the behaviour of a polynomiallyexact (e.g. Gaussian) cubature method. The proposed estimator has reduced mean square error compared to its competitors, and is illustrated on several Bayesian inference examples. All methods used in this paper are available in the R package ZVCV.

Markov chain Monte Carlo is the engine of modern Bayesian statistics, being used to approximate the posterior and derived quantities of interest. Despite this, the issue of how the output from a Markov chain is postprocessed and reported is often overlooked. Convergence diagnostics can be used to control bias via burn-in removal, but these do not account for (common) situations where a limited computational budget engenders a bias-variance trade-off. The aim of this article is to review state-of-the-art techniques for postprocessing Markov chain output. Our review covers methods based on discrepancy minimization, which directly address the bias-variance trade-off, as well as general-purpose control variate methods for approximating expected quantities of interest. Expected final online publication date for the Annual Review of Statistics and Its Application, Volume 9 is March 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

A novel control variate technique is proposed for post-processing of Markov chain Monte Carlo output, based both on Stein’s method and an approach to numerical integration due to Sard. The resulting estimators of posterior expected quantities of interest are proven to be polynomially exact in the Gaussian context, while empirical results suggest the estimators approximate a Gaussian cubature method near the Bernstein-von-Mises limit. The main theoretical result establishes a bias-correction property in settings where the Markov chain does not leave the posterior invariant. Empirical results are presented across a selection of Bayesian inference tasks. All methods used in this paper are available in the R package ZVCV.