On a normalized random measure with independent increments relevant to Bayesian nonparametric inference

Recently the class of normalized random measures with independent increments has been introduced. Such random probability measures, whose distributions act as nonparametric priors for Bayesian inference, are obtained by suitably normalizing time-changed increasing additive processes. We consider a particular normalized random measure with independent increments, which contains, as particular cases, the Dirichlet process, the normalized inverse Gaussian and stable random measures. Although its finite-dimensional distributions are not known, expressions for quantities of statistical interest can be derived. In particular, we provide simple rules for prior specification in terms of moments and obtain, in presence of exchangeable observations, its predictive distributions, which consist of a linear combination of the marginal distribution and of a weighted empirical distribution. We also study means of this random probability measure. Besides a necessary and sufficient condition for finiteness, we derive the exact prior and posterior distribution of any mean.