Random probability measures derived from increasing additive processes and their application to Bayesian statistics.

Increasing additive processes (IAP), i.e. processes with positive independent increments, represent a natural tool for defining random probability measures whose distributions act as nonparametric priors for Bayesian inference. It is well-known that the celebrated Dirichlet process can be obtained either by normalizing a time-changed gamma process or, as a particular case of neutral to the right (NTR) random probability measure, by the exponential transformation of a suitable IAP. A new class of random probability measures is introduced by generalizing the former construction to any IAP: a normalized random measure with independent increments (RMI) is defined by a suitable normalization of a time-changed IAP. Even if their finite-dimensional distributions are not generally known, quantities of statistical interest turn out to have, for a large subclass, appealing forms leading to simple rules for prior specification and to predictive distributions which consist of a linear combination of the marginal distribution and of a weighted empirical distribution. Particular attention is devoted to the study of their means. Necessary and sufficient conditions for finiteness together with their exact prior and approximate posterior distributions are provided. Some illustrative examples of statistical relevance are considered in detail. Normalized RMI can be further generalized to normalized IAP driven random measures, which contain the popular mixture of Dirichlet process as a particular case. Conditions for their existence are given. In particular, results for the distribution of means under both prior and posterior conditions are derived, and, relying upon the introduction of strategic latent variables, a full Bayesian analysis is undertaken. Moreover, NTR random probability measures are considered. Their means can be represented as the exponential functional of an IAP. This fact is exploited in order to give sufficient conditions for finiteness of the mean and for absolute continuity of its distribution. In addition, expressions for its moments, of any order, are provided. By resorting to the maximum entropy algorithm, an approximation to the density of the mean of a NTR prior is obtained. The numerical results are compared to those yielded by some well-established simulation algorithms in the context of a survival analysis problem