Search for: All records

Gaussian mixtures are commonly used for modeling heavy-tailed error distributions in robust linear regression. Combining the likelihood of a multivariate robust linear regression model with a standard improper prior distribution yields an analytically intractable posterior distribution that can be sampled using a data augmentation algorithm. When the response matrix has missing entries, there are unique challenges to the application and analysis of the convergence properties of the algorithm. Conditions for geometric ergodicity are provided when the incomplete data have a “monotone” structure. In the absence of a monotone structure, an intermediate imputation step is necessary for implementing the algorithm. In this case, we provide sufficient conditions for the algorithm to be Harris ergodic. Finally, we show that, when there is a monotone structure and intermediate imputation is unnecessary, intermediate imputation slows the convergence of the underlying Monte Carlo Markov chain, while post hoc imputation does not. An R package for the data augmentation algorithm is provided. 

Random-scan Gibbs samplers possess a natural hierarchical structure. The structure connects Gibbs samplers targeting higher-dimensional distributions to those targeting lower-dimensional ones. This leads to a quasi-telescoping property of their spectral gaps. Based on this property, we derive three new bounds on the spectral gaps and convergence rates of Gibbs samplers on general domains. The three bounds relate a chain’s spectral gap to, respectively, the correlation structure of the target distribution, a class of random walk chains, and a collection of influence matrices. Notably, one of our results generalizes the technique of spectral independence, which has received considerable attention for its success on finite domains, to general state spaces. We illustrate our methods through a sampler targeting the uniform distribution on a corner of an n-cube. 

Single-cell genomics has enabled the study of biological processes at an unprecedented scale and resolution. These studies were enabled by innovative data generation technologies coupled with emerging computational tools specialized for single-cell data. As single-cell technologies have become more prevalent, so has the development of new analysis tools, which has resulted in over 1,700 published algorithms1 (as of February 2024). Thus, there is an increasing need to continually evaluate which algorithm performs best in which context to inform best practices2,3 that evolve with the field. In many fields of quantitative science, public competitions and benchmarks address this need by evaluating state-of-the-art methods against known criteria, following the concept of a common task framework4. Here, we present Open Problems, a living, extensive, community-guided platform including 12 current single-cell tasks that we envisage raising standards for the selection, evaluation and development of methods in single-cell analysis.