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1. Estimating Convergence of Markov chains with L-Lag Couplings

   Biswas, Niloy; Jacob, Pierre E; Vanetti, Paul (January 2019, Advances in neural information processing systems)

   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. 

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2. Convergence speed and approximation accuracy of numerical MCMC

   https://doi.org/10.1017/apr.2024.28  

   Cui, Tiangang; Dong, Jing; Jasra, Ajay; Tong, Xin T (March 2025, Advances in Applied Probability)

   Abstract When implementing Markov Chain Monte Carlo (MCMC) algorithms, perturbation caused by numerical errors is sometimes inevitable. This paper studies how the perturbation of MCMC affects the convergence speed and approximation accuracy. Our results show that when the original Markov chain converges to stationarity fast enough and the perturbed transition kernel is a good approximation to the original transition kernel, the corresponding perturbed sampler has fast convergence speed and high approximation accuracy as well. Our convergence analysis is conducted under either the Wasserstein metric or the$$\chi^2$$metric, both are widely used in the literature. The results can be extended to obtain non-asymptotic error bounds for MCMC estimators. We demonstrate how to apply our convergence and approximation results to the analysis of specific sampling algorithms, including Random walk Metropolis, Metropolis adjusted Langevin algorithm with perturbed target densities, and parallel tempering Monte Carlo with perturbed densities. Finally, we present some simple numerical examples to verify our theoretical claims. 

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3. Distributed Bayesian Inference in Linear Mixed-Effects Models

   Srivastava, S; Xu, Y. (March 2021, Journal of computational and graphical statistics)

   null (Ed.)

   Linear mixed-effects models play a fundamental role in statistical methodology. A variety of Markov chain Monte Carlo (MCMC) algorithms exist for fitting these models, but they are inefficient in massive data settings because every iteration of any such MCMC algorithm passes through the full data. Many divide-and-conquer methods have been proposed to solve this problem, but they lack theoretical guarantees, impose restrictive assumptions, or have complex computational algorithms. Our focus is one such method called the Wasserstein Posterior (WASP), which has become popular due to its optimal theoretical properties under general assumptions. Unfortunately, practical implementation of the WASP either requires solving a complex linear program or is limited to one-dimensional parameters. The former method is inefficient and the latter method fails to capture the joint posterior dependence structure of multivariate parameters. We develop a new algorithm for computing the WASP of multivariate parameters that is easy to implement and is useful for computing the WASP in any model where the posterior distribution of parameter belongs to a location-scatter family of probability measures. The algorithm is introduced for linear mixed-effects models with both implementation details and theoretical properties. Our algorithm outperforms the current state-of-the-art method in inference on the functions of the covariance matrix of the random effects across diverse numerical comparisons. 

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4. Unbiased smoothing using particle independent Metropolis–Hastings

   Middleton, L.; Deligiannidis, G.; Doucet, A.; Jacob, P. E. (April 2019, AISTATS)

   We consider the approximation of expectations with respect to the distribution of a latent Markov process given noisy measurements. This is known as the smoothing problem and is often approached with particle and Markov chain Monte Carlo (MCMC) methods. These methods provide consistent but biased estimators when run for a finite time. We propose a simple way of coupling two MCMC chains built using Particle Independent Metropolis-Hastings (PIMH) to produce unbiased smoothing estimators. Unbiased estimators are appealing in the context of parallel computing, and facilitate the construction of confidence intervals. The proposed scheme only requires access to off-the-shelf Particle Filters (PF) and is thus easier to implement than recently proposed unbiased smoothers. The approach is demonstrated on a Lévy-driven stochastic volatility model and a stochastic kinetic model. 

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5. OPTIMAL AUXILIARY PRIORS AND REVERSIBLE JUMP PROPOSALS FOR A CLASS OF VARIABLE DIMENSION MODELS

   https://doi.org/10.1017/S0266466620000018  

   Norets, Andriy (February 2021, Econometric Theory)

   This article develops a Markov chain Monte Carlo (MCMC) method for a class of models that encompasses finite and countable mixtures of densities and mixtures of experts with a variable number of mixture components. The method is shown to maximize the expected probability of acceptance for cross-dimensional moves and to minimize the asymptotic variance of sample average estimators under certain restrictions. The method can be represented as a retrospective sampling algorithm with an optimal choice of auxiliary priors and as a reversible jump algorithm with optimal proposal distributions. The method is primarily motivated by and applied to a Bayesian nonparametric model for conditional densities based on mixtures of a variable number of experts. The mixture of experts model outperforms standard parametric and nonparametric alternatives in out of sample performance comparisons in an application to Engel curve estimation. The proposed MCMC algorithm makes estimation of this model practical. 

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