Conditional formulae for Gibbs-type exchangeable random partitions

Abstract

Gibbs-type random probability measures and the exchangeable random partitions they induce represent an important framework both from a theoretical and applied point of view. In the present paper, motivated by species sampling problems, we investigate some properties concerning the conditional distribution of the number of blocks with a certain frequency generated by Gibbs-type random partitions. The general results are then specialized to three noteworthy examples yielding completely explicit expressions of their distributions, moments and asymptotic behaviors. Such expressions can be interpreted as Bayesian nonparametric estimators of the rare species variety and their performance is tested on some real genomic data.Comment: Published in at http://dx.doi.org/10.1214/12-AAP843 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Posterior analysis of rare variants in Gibbs-type species sampling models

“…Our result generalizes Theorem 1 in Favaro et al [12], where the falling factorial moment of the random variable M l,n was derived. See also Ewens and Tavaré [10] for some moment formulae of M l,n under the framework of the Ewens sampling formula.…”

“…A particularly important example is represented by the estimation of the number of new species that will be observed in the additional sample. See Lijoi et al [29], Favaro et al [12], Favaro et al [11] and Bacallado et al [1] for estimators of other features related to species richness under Gibbs-type priors. This class of priors stands out for both mathematical tractability and flexibility.…”

“…Precisely, this is as a suitable convolution of: (i) the joint posterior distribution of the new rare variants that are generated from the additional sample and do not coincide with rare variants already detected in the initial sample; (ii) the joint posterior distribution of the old rare variants that arise by updating, via the additional sample, the rare variants already detected in the initial sample. Our distributional results are derived by generalizing some of the combinatorial techniques originally developed in Favaro et al [12], where special cases of the results in this paper have been presented. After submitting the first version of this paper we learnt that the posterior distribution of the rare variants have been recently obtained independently, and by means of different techniques, in Cerquetti [2].…”

“…Firstly, we devise a novel methodology to approximately quantify the uncertainty of a Bayesian nonparametric estimator for the number of rare species. This estimator has been recently introduced in Favaro et al [12] and the problem of evaluating its accuracy is of great importance in several applied contexts. To this end, we introduce approximate credible intervals that exploit the knowledge of the asymptotic behavior of the posterior distribution of rare variants.…”

“…These species will be referred to as old species. See Lijoi et al [29] and Favaro et al [12] for a description of the random variables (14)- (16) by means of conditional partition probability functions when data are generated by Gibbs-type priors.…”