Random Partition model and finitary Bayesian statistical inference

Exchangeability of observations corresponds to a condition shared by the vast majority of applications of the Bayesian paradigm. By de Finetti's rep- resentation theorem, if exchangeable observations form an infinite sequence of random variables, then they are conditionally independent and identi- cally distributed given some random parameter, which is the main object of statistical inference. Such parameter is a limiting mathematical entity and therefore hypotheses related to it might be not verifiable. For this reason, statistical analysis should be directed toward the prevision of the empirical distribution of N observations. In view of these considerations, specific forms of (finitary) exchangeable laws based on sequences of nested partitions have been introduced and studied in Bassetti and Bissiri (2007). In this paper, we intend to carry on this line of research studying another class of exchangeable laws, which rests on the concept of exchangeable random partition. These distributions are related to species sampling sequences, but allow negative correlation between observations. Marginal and predictive distributions are calculated together with the posterior distribution of the empirical process, and finally, it is shown how the predictive mean can be approximated by importance sampling.