Interpretations of probability often emphasize a frequentist approach, where probability is (only) a limiting frequency ratio. Another view is subjectivist, where probabilities are degrees of belief without necessarily having any physical manifestation.

Most existing interpretations for credal sets fall squarely
in the subjective regime (the same holds for related systems such
as belief functions, etc.). The fact that probabilities can
be directly related to observed frequencies gives probability a
significant advantage over other subjective
representations of belief. For example, as a result of this relationship,
decision analysts are often able to measure the *calibration* of an
expert's subjective assessments [16]. The lack of
a similar connection to observable physical outcomes for credal sets
is a troublesome deficiency for most existing theories. Only a few works
have attempted to make such connections, most notably the work of
Kyburg [13, 14], which proposes specific guidelines to
transform finite data knowledge into intervals of probabilities;
the work of Seidenfeld and Schervish [22]
on the convergence properties of beliefs in a group of agents;
and the work of Walley and Fine
[27] on estimators for sets of distributions.

Is it possible to relate a convex set of distributions to observable repeated outcomes in a manner analogous to the relationship between probabilities and frequencies? Can credal sets similarly be induced from a limiting series of observations in a meaningful fashion? Results by Walley and Fine [27] prove that such a connection is indeed possible. In this paper, we explain, build upon and extend these results, and we present interpretations of the mathematical results that are both useful and understandable.

With these results, interpretations of credal sets can, like interpretations of probability, have an additional grounding in observable phenomena, making notions such as calibration meaningful even for credal sets.

© Fabio Cozman[Send Mail?]

Sun Jun 29 22:16:40 EDT 1997