Minimax Predictive Density for Sparse Count Data

TL;DR

This paper introduces a class of Bayesian predictive densities optimized for high-dimensional sparse count data modeled by Poisson distributions, demonstrating asymptotic minimaxity and adaptivity, with practical applications and simulations.

Contribution

It develops asymptotically minimax and adaptive Bayesian predictive densities for sparse Poisson models, extending to quasi-sparsity and missing data scenarios.

Findings

Proposed predictive densities achieve asymptotic minimaxity.

Estimator with unknown sparsity level is adaptive.

Simulation and real data show efficiency of the method.

Abstract

This paper discusses predictive densities under the Kullback--Leibler loss for high-dimensional Poisson sequence models under sparsity constraints. Sparsity in count data implies zero-inflation. We present a class of Bayes predictive densities that attain 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 asymptotically 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 application to real data illustrate the efficiency of the proposed Bayes predictive densities.