**Sankhya:
The Indian Journal of Statistics**

2007, Volume 69, Pt. 4, 808--841

**Finitary Bayesian Statistical Inference Through
Partitions Tree Distributions**

Federico Bassetti, Universit\'a degli Studi di Pavia, Pavia, Italy

Pier Giovanni Bissiri, Universit\'a degli Studi di Cagliari, Cagliari, Italy

SUMMARY. According to the Bayesian theory, observations are usually considered to be part of an infinite sequence of random elements that are conditionally independent and identically distributed, given an unknown parameter. Such a parameter, which is the object of inference, depends on the entire sequence. Consequently, the unknown parameter cannot generally be observed, and any hypothesis about its realizations might be devoid of any empirical meaning. Therefore it becomes natural to focus attention on finite sequences of observations. The present paper introduces specific laws for finite exchangeable sequences and analyses some of their most relevant statistical properties. These laws, assessed through sequences of nested partitions, are strongly reminiscent of P\'olya-tree distributions and allow forms of conjugate analysis. As a matter of fact, this family of distributions, called partitions tree distributions, contains the exchangeable laws directed by the more familiar P\'olya-tree processes. Moreover, the paper gives an example of partitions tree distribution connected with the hypergeometric urn scheme, where negative correlation between past and future observations is allowed.

*AMS (2000) subject classification. *Primary 62C10, 62F15, 60G09.

*Key words and phrases. *de Finetti's theorem, empirical distribution, finitary Bayesian inference, finite exchangeability, hypergeometric distribution,
partitions tree distributions, P\'olya tree distributions, predictive inference,
random partitions, species sampling.