Finitary Bayesian statistical inference through partitions tree distributions

In Bayesian theory, observations are usually assumed to be part of an infinite sequence of random elements that are conditionally independent and identically distributed given some unknown parameter. Such a parameter, which is the object of inference, depends on the entire sequence. Consequently, it cannot generally be observed, and any hypothesis about the realizations of the unknown parameter might be devoid of any empirical meaning. This situation leads to ponder the advisability of directing statistical analysis toward the prevision of the empirical distribution of N observations or more generally toward functionals of such a distribution. According to this stance, it becomes natural to focus attentation on finite sequences of observations. In the present paper, specific laws for finite exchangeable sequences are given, and some of their most relevant statistical properties are studied. These laws, assessed through sequences of nested partitions, are strongly reminiscent of P?lya-tree distributions and allow forms of conjugate analysis. In fact, the class of these distributions, called partitions tree distributions contains the exchangeable laws directed by the more familiar P?lya-tree processes. Moreover, it encompasses an example of distribution connected with the hypergeometric urn scheme. It is worth stressing that, in this latter case, negative correlation between past and future observations are allowed