On Minimax Optimality of Sparse Bayes Predictive Density Estimates

2017/07/14 by Gourab Mukherjee, Iain M. Johnstone, Mukherjee, Gourab +1
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Bayesian Modeling and Causal Inference #FOS: Mathematics #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.1707.04380

openalex publication_date 2017/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study predictive density estimation under Kullback-Leibler loss in ℓ₀-sparse Gaussian sequence models. We propose proper Bayes predictive density estimates and establish asymptotic minimaxity in sparse models. A surprise is the existence of a phase transition in the future-to-past variance ratio r. For r < r₀ = (\surd 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-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, spike-and-slab priors can attain asymptotic minimaxity.

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