Introduction to Imprecise Probabilities

1.6 Relationships with other, nonequivalent models

We have studied sets of desirable gambles and investigated their connection to partial preference orders, which gave rise to strict and nonstrict variants. In the preceding section we settled on the strict variant. Sets of desirable gambles and partial preference orders are equivalent uncertainty models: one can be expressed in terms of the other and vice versa. In this section, we investigate their connection with other, commonly used, but nonequivalent models.

Given an assessment denoted , where is a dummy variable that stands for an uncertainty model generating the assessment, we will use the notational conventions and .

1.6.1 Linear previsions

Linear previsions are positive, linear, normed functionals and popular uncertainty models in classical probability theory. A linear prevision provides fair prices for gambles, i.e., it is a real-valued function on . (We imbue the set of real-valuedfunctionals on with the topology of pointwise ...