A note on Bayesian nonparametric priors derived from exponentially tilted Poisson–Kingman models

We derive the class of normalized generalized Gamma processes from Pitman's Poisson–Kingman models [Pitman, J., 2003. Poisson–Kingman partitions. In: Goldstein, D.R., (Ed.), Science and Statistics: A Festschrift for Terry Speed, IMS Lecture Notes–Monograph Series, vol. 40. Institute of Mathematical Statistics, Hayward, CA, pp. 1–34.] with tempered αα-stable mixing distribution. Relying on this construction it can be shown that in Bayesian nonparametrics, results on quantities of statistical interest under those priors, like the analogous of the Blackwell–MacQueen prediction rules or the distribution of the number of distinct elements observed in a sample, arise as immediate consequences of Pitman's results.