ON MINIMAX OPTIMALITY OF SPARSE BAYES PREDICTIVE DENSITY ESTIMATES

• 成果类型: Article
• 署名作者: Mukherjee, Gourab; Johnstone, Iain M.
• 署名单位: University of Southern California; Stanford University; Stanford University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/21-AOS2086
• 发表日期: 2022
• 页码: 81-106
• 关键词: VARIABLE SELECTION; location families; shrinkage priors; spike; needles; straw

摘要

We study predictive density estimation under Kullback-Leibler loss in l(0)-sparse Gaussian sequence models. We propose proper Bayes predictive density estimates and establish asymptotic minimaxity in sparse models. Fundamental for this is a new risk decomposition for sparse, or spike-and-slab priors. A surprise is the existence of a phase transition in the future-to-past variance ratio r. For r < r(0) = (root 5 - 1)/4, the natural discrete prior ceases to be asymptotically optimal. Instead, for subcritical r, a 'bi-grid' prior with a central region of reduced grid spacing recovers asymptotic minimaxity. This phenomenon seems to have no analog in the otherwise parallel theory of point estimation of a multivariate normal mean under quadratic loss. For spike-and-uniform slab priors to have any prospect of minimaxity, we show that the sparse parameter space needs also to be magnitude constrained. Within a substantial range of magnitudes, such spike-and-slab priors can attain asymptotic minimaxity.