We show that for any correlated multiple-try Metropolis algorithm, one can always accept the proposal and evaluate the importance weight that is needed to correct for the bias without extra computational cost. This results in a general class of convenient and rejection-free Markov chain Monte Carlo (MCMC) sampling schemes. By further leveraging the importance sampling perspective on Metropolis–Hastings algorithms, we propose an alternative MCMC sampler on discrete spaces along with a general theory on its complexity. Both proposed methods can be viewed as using importance weighting to overcome the inefficiency of the Metropolis–Hastings framework caused by repeated proposal rejections. Numerical examples suggest that the proposed algorithms are consistently more efficient than their Metropolis–Hastings counterparts.
High-dimensional interval-censored survival data are increasingly encountered in fields such as biomedical research, finance, and the social sciences, presenting substantial challenges for variable selection and parameter estimation. In this paper, we introduce a novel Skinny Variational Bayesian (SVB) algorithm designed for efficient parametric inference in high-dimensional Cox proportional hazards models with interval-censored outcomes. The proposed method combines the Spike-and-Slab sparse prior with the computational efficiency of the Skinny Gibbs sampler, striking an effective balance between scalability and the inferential rigor of Bayesian methodology. Extensive simulation studies demonstrate that the Skinny VB offers a scalable balance between estimation accuracy, signal recovery, and computational efficiency. The utility of the proposed approach is further validated through application to data from an HIV vaccine trial, where it successfully identifies key immunological biomarkers associated with HIV-1 infection risk and disease progression.
The scalability of stochastic gradient MCMC algorithms has made them increasingly important for Bayesian analysis of big data. However, these algorithms are prone to getting trapped in local energy minima. We introduce a novel algorithm—evolutionary stochastic gradient Langevin dynamics (ESGLD)— that is particularly effective at mitigating local-trap behavior when simulating posterior distributions in big data scenarios. ESGLD operates with a population of Markov chains, each at a different temperature, and these chains interact through a snooker operator. The snooker operator performs a short run of SGLD along the line connecting the current sample to a selected anchor point. As long as a mode is located by other chains in the population, the target chain can transition to it by running the snooker operator, particularly when the mode is sharp and well-separated from others. The snooker operator enables the Markov chain to effectively escape local traps while maintaining invariance with respect to the target distribution. This significantly accelerates convergence, especially when the energy landscape is rugged. We provide theoretical guarantees for the convergence of the ESGLD algorithm in the big data regime and demonstrate its numerical effectiveness through simulated and real data examples, with the latter modeled by neural networks.
Group testing is an efficient means of screening for infectious disease that combines specimens from multiple individuals and tests them as a single ‘pooled’ specimen. If a pool tests positive, subsequent tests are usually conducted to determine the disease status of the contributing individuals; if a pool tests negative, all samples in the pool are considered disease free. In recent years, Bayesian regression frameworks for group testing data have been developed, offering flexibility across group testing protocols and enabling the estimation of unknown assay accuracies. However, existing Bayesian methods rely heavily on Markov chain Monte Carlo (MCMC) sampling for estimation and inference, which becomes increasingly computationally inefficient as sample sizes grow, creating a bottleneck that limits the applicability of these models to large datasets. Variational Bayes (VB), an optimization-based algorithm that provides an analytical approximation to the posterior distribution, offers a more computationally efficient alternative to MCMC. To overcome the computational challenges of current Bayesian group testing methods, we develop a VB algorithm for group testing regression. Because VB methods are known to underestimate the posterior variance, we also propose an alternative inference method that exploits the asymptotic normality of VB estimators. We demonstrate with numerical studies that this hybrid inference approach achieves comparable inference to the MCMC approach while being more computationally scalable. We then apply the proposed VB method to COVID-19 group testing data from 43,185 individuals in the National Institutes of Health’s All of Us Research Program.
Modeling multivariate discrete or ordinal variables poses non-trivial challenges. Nevertheless, latent variable models, particularly those relying on the multivariate normal distribution, have been extensively studied and widely used for this purpose. In this work, we develop a modeling framework for multivariate discrete and ordinal variables constructed via a Student’s t latent distribution. Focusing in particular on the bivariate case, we derive expressions for the joint probabilities of discretized variables and propose computational procedures that enable their accurate and efficient evaluation, which are crucial for inferential purposes, where repeated likelihood evaluations are required. Building on this computational framework, we provide a systematic analysis of the mapping between the copula dependence parameters (correlation and degrees of freedom) and the resulting linear correlation of ordinal variables. Particular attention is devoted to the number of categories and to distributional shape, focusing on four prototypical types of discrete distributions: uniform, decreasing, symmetric reverse U-shaped, and symmetric U-shaped. We also develop a numerical optimization algorithm that identifies the marginal distributions that maximize the resulting correlation for a given t correlation, degrees of freedom, and fixed number of ordinal categories. A key finding is that discretization can inflate the observed correlation when degrees of freedom are small and margins are highly asymmetric, in contrast with the Gaussian copula, where discretization always attenuates correlation. Estimation issues are addressed through both a two-step and a full maximum likelihood procedure for the proposed discrete model. Their statistical and computational performance is evaluated and compared numerically. Finally, we provide an R implementation to support and illustrate the proposed methods, with applications to real datasets.
