On minimax optimality of sparse Bayes predictive density estimates

Abstract

We study predictive density estimation under Kullback–Leibler loss in ${\ell }_0$-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 $\mathit{r}<{\mathit{r}}_0=(\surd 5-1)/4$, 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-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.