Scaled process priors for Bayesian nonparametric estimation of the unseen genetic variation

There is a growing interest in the estimation of the number of unseen features, mostly driven by applications in biological sciences. A recent work brought out the upside and the downside of the popular stable-Beta process prior, and generalizations thereof, in Bayesian nonparametric inference for the unseen-features problem: i) the downside lies in the limited use of the sampling information in the posterior distributions, which depend on the observable sample only through the sample size; ii) the upside lies in the analytical tractability and interpretability of the posterior distributions, which are simple Poisson distributions whose parameters are simple to compute, and depend on the sample size and the prior's parameter. In this paper, we introduce and investigate an alternative nonparametric prior, referred to as the stable-Beta scaled process prior, which is the first prior that allows to enrich the posterior distribution of the number of unseen features, through the inclusion of the sampling information on the number of distinct features in the observable sample, while maintaining the same analytical tractability and interpretability as the stable-Beta process prior. Our prior leads to a negative Binomial posterior distribution, whose parameters depends on the sample size, the observed number of distinct features and the prior's parameter, providing estimates that are simple, linear in the sampling information and computationally efficient. We apply our approach to synthetic and real genetic data, showing that it outperforms parametric and nonparametric competitors in terms of estimation accuracy.