On minimax optimality of sparse Bayes predictive density estimates
研究了在Kullback-Leibler损失下，稀疏高斯序列模型中贝叶斯预测密度估计的渐近极小极大性，发现未来与过去方差比存在相变，并提出了新的先验设计。
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.