Bayesian inference with the L1-ball prior: solving combinatorial problems with exact zeros

• 成果类型: Article
• 署名作者: Xu, Maoran; Duan, Leo L.
• 署名单位: Duke University; State University System of Florida; University of Florida; State University System of Florida; University of Florida
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412
• DOI: 10.1093/jrsssb/qkad076
• 发表日期: 2024
• 页码: 1538-1560
• 关键词: VARIABLE SELECTION; DECOMPOSITION; shrinkage; mixtures; monotone; models; number; Lasso; NORM

摘要

The l(1)-regularisation is very popular in high-dimensional statistics-it changes a combinatorial problem of choosing which subset of the parameter is zero, into a simple continuous optimisation. Using a continuous prior concentrated near zero, the Bayesian counterparts are successful in quantifying the uncertainty in the variable selection problems; nevertheless, the lack of exact zeros makes it difficult for broader problems such as change-point detection and rank selection. Inspired by the duality of the l(1)-regularisation as a constraint onto an l(1)-ball, we propose a new prior by projecting a continuous distribution onto the l(1)-ball. This creates a positive probability on the ball boundary, which contains both continuous elements and exact zeros. Unlike the spike-and-slab prior, this l(1)-ball projection is continuous and differentiable almost surely, making the posterior estimation amenable to the Hamiltonian Monte Carlo algorithm. We examine the properties, such as the volume change due to the projection, the connection to the combinatorial prior, the minimax concentration rate in the linear problem. We demonstrate the usefulness of exact zeros that simplify the combinatorial problems, such as the change-point detection in time series, the dimension selection of mixture models, and the low-rank plus-sparse change detection in medical images.