Abstract
In Bayesian nonparametric statistics, it is crucial that the support of the prior is very large. Here, we consider species sampling priors. Such priors are widely used within mixture models and it has been shown in the literature that a large support for the mixing prior is essential to ensure the consistency of the posterior. In this paper, simple conditions are given that are necessary and sufficient for the support of a species sampling prior to be full. In particular, for proper species sampling priors, the condition is that the maximum size of the atoms of the corresponding process is small with positive probability. We apply this result to show that the main classes of species sampling priors known in literature have full support under mild conditions. Moreover, we find priors with a very simple construction still having full support.
Highlights
The main motive of Bayesian nonparametrics is to avoid restrictive parametric assumptions about the distribution generating the data
This paper considers the support of the distribution of species sampling processes, which are a relevant class of discrete random probability measures
Species sampling models, which have been developed by Pitman [31], Hansen and Pitman [14], have been extensively studied in Bayesian nonparametric statistical literature
Summary
The main motive of Bayesian nonparametrics is to avoid restrictive parametric assumptions about the distribution generating the data This is done constructing instead random probability measures whose distributions to be used as priors have a large support (see, for instance, Ghosal [10]). This paper considers the support (with respect to the weak topology) of the distribution of species sampling processes, which are a relevant class of discrete random probability measures. By means of our results, we shall find priors with full support that have a very simple construction and are expected to be easier to implement than the more complex Bayesian nonparametric models This is the direction pointed by Fuentes-Garcıa, Mena and Walker [9], who consider the species sampling prior with geometric weights. The Appendix contains the proof of Theorem 2 and Proposition 6
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