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 or quasi zero-inflation, that is, situations where there exists an excess of zeros or near-zero counts. We investigate the exact asymptotically minimax Kullback--Leibler risks in sparse and quasi-sparse Poisson sequence models. We also provide a class of Bayes predictive densities that attain exact asymptotic minimaxity. To this end, we employ spike-and-slab priors with improper slab priors and introduce the notion of predictive specification of the scale in the mixture of improper priors. For application, we provide exact asymptotically minimax predictive densities that are adaptive to an unknown sparsity, and discuss the performance of the proposed Bayes predictive densities in settings where current observations are missing completely at random. The simulation studies as well as applications to real data demonstrate the efficiency of the proposed Bayes predictive densities.