On the problem of computing the conglomerable natural extension

Abstract

Embedding conglomerability as a rationality requirement in probability was among the aims of Walley's behavioural theory of coherent lower previsions. However, recent work has shown that this attempt has only been partly successful. If we focus in particular on the extension of given assessments to a rational and conglomerable model (in the least-committal way), we have that the procedure used in Walley's theory, the natural extension, provides only an approximation to the model that is actually sought for: the so-called conglomerable natural extension. In this paper we consider probabilistic assessments in the form of a coherent lower prevision P̲, which is another name for a lower expectation functional, and make an in-depth mathematical study of the problem of computing the conglomerable natural extension for this case: that is, where it is defined as the smallest coherent lower prevision F̲≥P̲ that is conglomerable, in case it exists. Past work has shown that F̲ can be approximated by an increasing sequence (E̲n)n∈N of coherent lower previsions. We solve an open problem by showing that this sequence can consist of infinitely many distinct elements. Moreover, we give sufficient conditions, of quite broad applicability, to make sure that the point-wise limit of the sequence is F̲ in case P̲ is the lower envelope of finitely many linear previsions. In addition, we study the question of the existence of F̲ and its relationship with the notion of marginal extension.