A Good-Turing estimator for feature allocation models
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Feature allocation models generalize 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, given n samples, we study the problem of estimating the missing mass M_n, namely the expected number hitherto unseen features that would be observed if one additional individual was sampled. This 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 introduce a simple, robust and theoretically sound nonparametric estimator M̂_n of M_n. M̂_n turns out to have the same analytic form of the popular Good-Turing estimator of the missing mass in species sampling models, with the difference that the two estimators have different ranges. We show that M̂_n admits a natural interpretation both as a jackknife estimator and as a nonparametric empirical Bayes estimator, we give provable guarantees for the performance of M̂_n in terms of minimax rate optimality, and we provide with an interesting connection between M̂_n and the Good-Turing estimator for species sampling. Finally, we derive non-asymptotic confidence intervals for M̂_n, 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.
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