A good-turing estimator for feature allocation models

�Feature allocation models generalize classical species sampling models by allowing every observation to belong to more than one species, now called features. Under the popular Bernoulli product model for feature allocation, we assume n observable samples and we consider the problem of estimating the expected number Mn of hitherto unseen features that would be observed if one additional individual was sampled. The interest in estimating Mn is motivated by numerous applied problems where the sampling procedure is expensive, in terms of time and/or financial resources allocated, and further samples can be only motivated by the possibility of recording new unobserved features. We consider a nonparametric estimator ˆMn of Mn which has the same analytic form of the popular Good-Turing estimator of the missing mass in the context of species sampling models. We show thatˆMn admits a natural interpretation both as a jackknife estimator and as a nonparametric empirical Bayes estimator. Furthermore, we give provable guarantees for the performance of ˆMn in terms of minimax rate optimality, and we provide with an interesting connection betweenˆMn and the Good-Turing estimator for species sampling. Finally, we derive non-asymptotic confidence intervals forˆMn, which are easily computable and do not rely on any asymptotic approximation. Our approach is illustrated with synthetic data and SNP data from the ENCODE sequencing genome project.