Statistical issues connected with finitary exchangeable sequences

Exchangeability of observations corresponds to a condition shared by the vast majority of applications of the Bayesian paradigm. By de Finetti's representation theorem, if exchangeable observations form an infinite sequence of random variables, then they are conditionally independent and identically 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, two specific forms of (finitary) exchangeable laws are introduced and studied: one is based on sequences of nested partitions and the other one rests on the concept of exchangeable random partition.