Sparse Minimax Optimality of Bayes Predictive Density Estimates from Clustered Discrete Priors

Abstract

We consider the problem of predictive density estimation under Kullback-Leibler loss in a high-dimensional Gaussian model with exact sparsity constraints on the location parameters. We study the first order asymptotic minimax risk of Bayes predictive density estimates based on product discrete priors where the proportion of non-zero coordinates converges to zero as dimension increases. Discrete priors that are product of clustered univariate priors provide a tractable configuration for diversification of the future risk and are used for constructing efficient predictive density estimates. We establish that the Bayes predictive density estimate from an appropriately designed clustered discrete prior is asymptotically minimax optimal. The marginals of our proposed prior have infinite clusters of identical sizes. The within cluster support points are equi-probable and the clusters are periodically spaced with geometrically decaying probabilities as they move away from the origin. The cluster periodicity depends on the decay rate of the cluster probabilities. Under different sparsity regimes, through numerical experiments, we compare the maximal risk of the Bayes predictive density estimates from the clustered prior with varied competing estimators including those based on geometrically decaying non-clustered priors of Johnstone (1994) and Mukherjee & Johnstone (2017) and obtain encouraging results.