Bayesian hierarchical models are powerful tools for learning common latent features across multiple data sources. The Hierarchical Dirichlet Process (HDP) is invoked when the number of latent components is a priori unknown. While there is a rich literature on finite sample properties and performance of hierarchical processes, the analysis of their frequentist posterior asymptotic properties is still at an early stage. Here we establish theoretical guarantees for recovering the true data generating process when the data are modeled as mixtures over the HDP or a generalization of the HDP, which we term boosted because of the faster growth in the number of discovered latent features. By extending Schwartz's theory to partially exchangeable sequences we show that posterior contraction rates are crucially affected by the relationship between the sample sizes corresponding to the different groups. The effect varies according to the smoothness level of the true data distributions. In the supersmooth case, when the generating densities are Gaussian mixtures, we recover the parametric rate up to a logarithmic factor, provided that the sample sizes are related in a polynomial fashion. Under ordinary smoothness assumptions more caution is needed as a polynomial deviation in the sample sizes could drastically deteriorate the convergence to the truth.

Discrete random probability measures and the exchangeable random partitions they induce are key tools for addressing a variety of estimation and prediction problems in Bayesian inference. Here we focus on the family of Gibbs-type priors, a recent elegant generalization of the Dirichlet and the Pitman-Yor process priors. These random probability measures share properties that are appealing both from a theoretical and an applied point of view: (i) they admit an intuitive predictive characterization justifying their use in terms of a precise assumption on the learning mechanism; (ii) they stand out in terms of mathematical tractability; (iii) they include several interesting special cases besides the Dirichlet and the Pitman-Yor processes. The goal of our paper is to provide a systematic and unified treatment of Gibbs-type priors and highlight their implications for Bayesian nonparametric inference. We deal with their distributional properties, the resulting estimators, frequentist asymptotic validation and the construction of time-dependent versions. Applications, mainly concerning mixture models and species sampling, serve to convey the main ideas. The intuition inherent to this class of priors and the neat results they lead to make one wonder whether it actually represents the most natural generalization of the Dirichlet process.

BACKGROUND: Expressed sequence tags (ESTs) analyses are a fundamental tool for gene identification in organisms. Given a preliminary EST sample from a certain library, several statistical prediction problems arise. In particular, it is of interest to estimate how many new genes can be detected in a future EST sample of given size and also to determine the gene discovery rate: these estimates represent the basis for deciding whether to proceed sequencing the library and, in case of a positive decision, a guideline for selecting the size of the new sample. Such information is also useful for establishing sequencing efficiency in experimental design and for measuring the degree of redundancy of an EST library. RESULTS: In this work we propose a Bayesian nonparametric approach for tackling statistical problems related to EST surveys. In particular, we provide estimates for: a) the coverage, defined as the proportion of unique genes in the library represented in the given sample of reads; b) the number of new unique genes to be observed in a future sample; c) the discovery rate of new genes as a function of the future sample size. The Bayesian nonparametric model we adopt conveys, in a statistically rigorous way, the available information into prediction. Our proposal has appealing properties over frequentist nonparametric methods, which become unstable when prediction is required for large future samples. EST libraries, previously studied with frequentist methods, are analyzed in detail. CONCLUSION: The Bayesian nonparametric approach we undertake yields valuable tools for gene capture and prediction in EST libraries. The estimators we obtain do not feature the kind of drawbacks associated with frequentist estimators and are reliable for any size of the additional sample.