Specification of the base measure of nonparametric priors via random means

Functionals of random probability measures are probabilistic objects whose properties are studied in different fields. They also play an important role in Bayesian Nonparametrics: understanding the behavior of a finite dimensional feature of a flexible and infinite-dimensional prior is crucial for prior elicitation. In particular distributions of means of nonparametric priors have been the object of thorough investigation in the literature. We target the inverse path: the determination of the parameter measure of a random probability measure giving rise to a fixed mean distribution. This direction yields a better understanding of the sets of mean distributions of notable nonparametric priors, giving moreover a way to directly enforce prior information, without losing inferential power. Here we summarize and report results obtained in [6] for the Dirichlet process, the normalized stable random measure and the Pitman–Yor process, with an application to mixture models

Two‐group Poisson‐Dirichlet mixtures for multiple testing

The simultaneous testing of multiple hypotheses is common to the analysis of high-dimensional data sets. The two-group model, first proposed in Efron (2004), identifies significant comparisons by allocating observations to a mixture of an empirical null and an alternative distribution. In the Bayesian nonparametrics literature, many approaches have suggested using mixtures of Dirichlet Processes in the two group model framework. Here, we investigate employing instead mixtures of two-parameter Poisson Dirichlet Processes (2PPD), and show how they provide a more flexible and effective tool for large-scale hypothesis testing. Our model further employs non-local prior densities to allow separation between the two mixture components. We obtain a closed form expression for the exchangeable partition probability function of the two-group model, which leads to a straightforward MCMC implementation. We compare the performances of our method for large-scale inference in a simulation study and illustrate its use on both a prostate cancer dataset and a case-control microbiome study of the gastrointestinal tracts in children from underdeveloped countries who have been recently diagnosed with moderate to severe diarrhe