Exchangeability and Data Analysis

Abstract

The term 'exchangeability' was introduced by de Finetti in the context of personal probability to describe a particular sense in which quantities treated as random variables in a probability specification are thought to be similar. We believe, however, that judgments of similarity are more primitive than those of probability and are at the heart of all statistical activities, including those for which probability specifications are absent or contrived. In this paper, we give a definition of exchangeability in a descriptive context, which extends de Finetti's concept to a wider domain. Our objective is to analyse the logic of judgments of exchangeability (or similarity, or homogeneity), to clarify the roles of context and data analysis in these judgments. We give several examples to illustrate the nature of these judgments in description, inference and prediction. We use this discussion to clarify the extent to which judgments of similarity in inference and prediction can be based on data, and the extent to which they must rely on pure faith. Our discussion is a contribution to the emerging theory of data analysis, the as yet largely atheoretical and informal process that precedes and supports formal statistical activities.