On minimax optimality of sparse Bayes predictive density estimates
Gourab Mukherjee, Iain M. Johnstone
The Annals of Statistics · 2022 · Volume 50 · Issue 1
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. 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<r0=(√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.
Keywords
62C12, 62C25, 62F10, 62J07, asymptotic minimaxity, high dimensional, least favorable prior, predictive density, proper Bayes rule, Sparsity, spike and slab
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