Contaminated Gibbs-Type Priors

Gibbs-type priors are combinatorial processes widely used as key components in several Bayesian nonparametric models. By virtue of their flexibility and mathematical tractability, they turn out to be predominant priors in species sampling problems and mixture modeling. We introduce a new family of processes which extends the Gibbs-type one, by including a contaminant component in the model to account for an excess of observations with frequency one. We first investigate the induced random partition, the associated predictive distribution, the asymptotic behavior of the total number of blocks and the number of blocks with a given frequency: all the results we obtain are in closed form and easily interpretable. A remarkable aspect of contaminated Gibbs-type priors relies on their predictive structure, compared to the one of the standard Gibbs-type family: it depends on the additional sampling information on the number of observations with frequency one out of the observed sample. As a noteworthy example we focus on the contaminated version of the Pitman-Yor process, which turns out to be analytically tractable and computationally feasible. Finally we pinpoint the advantage of our construction in different applications: we show how it helps to improve predictive inference in a species-related dataset exhibiting a high number of species with frequency one; we also discuss the use of the proposed construction in mixture models to perform density estimation and outlier detection.