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Spike-and-slab priors are commonly used for Bayesian variable selection, due to their interpretability and favorable statistical properties. However, existing samplers for spike-and-slab posteriors incur prohibitive computational costs when the number of variables is large. In this article, we propose Scalable Spike-and-Slab (S^3), a scalable Gibbs sampling implementation for high-dimensional Bayesian regression with the continuous spike-and-slab prior of George & McCulloch (1993). For a dataset with n observations and p covariates, S^3 has order max{n^2 p_t, np} computational cost at iteration t where p_t never exceeds the number of covariates switching spike-and-slab states between iterations t and t-1 of the Markov chain. This improves upon the order n^2 p per-iteration cost of state-of-the-art implementations as, typically, p_t is substantially smaller than p. We apply S^3 on synthetic and real-world datasets, demonstrating orders of magnitude speed-ups over existing exact samplers and significant gains in inferential quality over approximate samplers with comparable cost. 

Statisticians often use Monte Carlo methods to approximate probability distributions, primarily with Markov chain Monte Carlo and importance sampling. Sequential Monte Carlo samplers are a class of algorithms that combine both techniques to approximate distributions of interest and their normalizing constants. These samplers originate from particle filtering for state space models and have become general and scalable sampling techniques. This article describes sequential Monte Carlo samplers and their possible implementations, arguing that they remain under-used in statistics, despite their ability to perform sequential inference and to leverage parallel processing resources among other potential benefits. Supplementary materials for this article are available online. 

We present a Gibbs sampler for the Dempster–Shafer (DS) approach to statistical inference for categorical distributions. The DS framework extends the Bayesian approach, allows in particular the use of partial prior information, and yields three-valued uncertainty assessments representing probabilities “for,” “against,” and “don’t know” about formal assertions of interest. The proposed algorithm targets the distribution of a class of random convex polytopes which encapsulate the DS inference. The sampler relies on an equivalence between the iterative constraints of the vertex configuration and the nonnegativity of cycles in a fully connected directed graph. Illustrations include the testing of independence in 2 × 2 contingency tables and parameter estimation of the linkage model. 

Couplings play a central role in the analysis of Markov chain Monte Carlo algorithms and appear increasingly often in the algorithms themselves, e.g. in convergence diagnostics, parallelization, and variance reduction techniques. Existing couplings of the Metropolis-Hastings algorithm handle the proposal and acceptance steps separately and fall short of the upper bound on one-step meeting probabilities given by the coupling inequality. This paper introduces maximal couplings which achieve this bound while retaining the practical advantages of current methods. We consider the properties of these couplings and examine their behavior on a selection of numerical examples. 

Markov chain Monte Carlo (MCMC) methods generate samples that are asymptotically distributed from a target distribution of interest as the number of iterations goes to infinity. Various theoretical results provide upper bounds on the distance between the target and marginal distribution after a fixed number of iterations. These upper bounds are on a case by case basis and typically involve intractable quantities, which limits their use for practitioners. We introduce L-lag couplings to generate computable, non-asymptotic upper bound estimates for the total variation or the Wasserstein distance of general Markov chains. We apply L-lag couplings to the tasks of (i) determining MCMC burn-in, (ii) comparing different MCMC algorithms with the same target, and (iii) comparing exact and approximate MCMC. Lastly, we (iv) assess the bias of sequential Monte Carlo and self-normalized importance samplers.