Exact Minimax Predictive Density for Sparse Count Data

This paper discusses predictive densities under the Kullback--Leibler loss in high-dimensional sparse count data models. In particular, Poisson sequence models under sparsity constraints are discussed. Sparsity in count data implies zero-inflation. We present a class of Bayes predictive densities that attain exact asymptotic minimaxity in sparse Poisson sequence models. We also show that our class with an estimator of unknown sparsity level plugged-in is adaptive in the exact minimax sense. For application, we extend our results to settings with quasi-sparsity and with missing-completely-at-random observations. The simulation studies as well as applications to real data demonstrate the efficiency of the proposed Bayes predictive densities.