Characterization of the law of a finite exchangeable sequence through the finite-dimensional distributions of the empirical measure
Introduction
Statisticians usually deal with observations whose order is deemed irrelevant. This idea of symmetry is translated into probabilistic terms by resorting to the notion of exchangeability, as used and studied by Bruno de Finetti.
If observations form an infinite sequence of exchangeable random elements (and take values into a Polish space), then they are conditionally independent and identically distributed, given a random probability measure (de Finetti’s representation theorem of infinite exchangeable sequences). The law of is unique and is called the de Finetti’s measure.
Methods to characterize laws of infinite exchangeable sequences are found in the literature. Regazzini and Petris (1992) and Regazzini (2001) single out conditions being necessary and sufficient so that a family of finite-dimensional distributions provides a complete characterization of de Finetti’s measure. Moreover, Fortini et al. (2000) show how a sequence of predictive distributions can be assessed as being consistent with an infinite exchangeable sequence.
In most of the literature about exchangeability, the observation process is assumed to be indefinitely long and therefore only infinite sequences are considered. Finite exchangeable sequences have not generally received much attention, although real statistical problems generally arise either in connection with finite sets of trials or in connection with finite populations.
If observations are assumed to form a finite exchangeable sequence , then de Finetti’s conditional representation need not be satisfied. However, one can resort to a finite version of de Finetti’s theorem (due to de Finetti, as well), where the usual directing measure is replaced by the empirical measure This theorem states that a finite sequence of random elements is exchangeable if and only if, for each , conditionally on , the random elements are distributed as drawings without replacement from an urn with balls, among which balls are labeled as “”, for each atom of . Therefore, the law of a finite exchangeable sequence is determined by the specification of the probability distribution of its empirical measure. This and other peculiarities of finite exchangeable sequences can be found, for instance, in Kingman, 1978c, Diaconis and Freedman, 1980, Aldous, 1985, Schervish, 1995, Spizzichino, 1982 and Wood (1992).
The aim of this paper is to identify the law of a finite exchangeable sequence through the finite-dimensional distributions of the empirical measure, i.e. the laws of the random vectors , where is a -tuple of measurable subsets of the range of the single observation. The problem will be approached in Section 1, singling out conditions that are necessary and sufficient so that a family of finite-dimensional distributions provides a complete characterization of the law of the empirical measure.
In Section 2, this result is applied to assess the law of a finite exchangeable sequence through the empirical measure. First, the section considers a construction that rests on the concept of random partition and yields the Random Partition Model introduced and studied by Bassetti and Bissiri (2008). Then, it is assumed that each observation takes value into a Polish space and a method is developed that is based on a nested sequence of partitions and allows to define the partitions tree distributions introduced by Bassetti and Bissiri (2007).
Section snippets
The main result
Let be a finite sequence of observations. Such observations may refer to the values that a character of interest takes on the units of a finite population or to the results in a given experiment. In the latter case, the value is the maximum number of trials that can be performed.
Suppose that each observation takes values into a measurable space . Each observation can be viewed as a measurable function from into according to the following definition:
Assessment of the law of a finite exchangeable sequence
The present section shows how to apply the Theorem 2 in order to concretely assess the law of a finite exchangeable sequence through the empirical measure. First, a construction is considered that is based on exchangeable random partitions and yields the Random Partition Model introduced by Bassetti and Bissiri (2008). Then, it is assumed that each observation takes value into a Polish space and another construction is presented that is based on a nested sequence of partitions and allows us to
Acknowledgements
The author is grateful to Eugenio Regazzini for providing much of the inspiration behind this paper, to Piercesare Secchi for his encouragement and to Stephen G. Walker for his helpful suggestions and his encouragement. Moreover, he wishes to thank an anonymous referee for helpful comments on an earlier version of the paper.
This work was partially supported by ESF and Regione Lombardia (by the grant “Dote Ricercatori”).
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