In mathematics, a credal set is a set of probability distributions[1] or, more generally, a set of (possibly only finitely additive) probability measures. A credal set is often assumed or constructed to be a closed convex set. It is intended to express uncertainty or doubt about the probability model that should be used, or to convey the beliefs of a Bayesian agent about the possible states of the world.[2]
If a credal set is closed and convex, then, by the Krein–Milman theorem, it can be equivalently described by its extreme points . In that case, the expectation for a function of with respect to the credal set forms a closed interval , whose lower bound is called the lower prevision of , and whose upper bound is called the upper prevision of :[3]
where denotes a probability measure, and with a similar expression for (just replace by in the above expression).
If is a categorical variable, then the credal set can be considered as a set of probability mass functions over .[4]
If additionally is also closed and convex, then the lower prevision of a function of can be simply evaluated as:
where denotes a probability mass function.
It is easy to see that a credal set over a Boolean variable cannot have more than two extreme points (because the only closed convex sets in are closed intervals), while credal sets over variables that can take three or more values can have any arbitrary number of extreme points.[citation needed]
^Levi, I. (1980). The Enterprise of Knowledge. MIT Press, Cambridge, Massachusetts.
^Cozman, F. (1999). Theory of Sets of Probabilities (and related models) in a Nutshell Archived 2011-07-21 at the Wayback Machine.
^Walley, Peter (1991). Statistical Reasoning with Imprecise Probabilities. London: Chapman and Hall. ISBN 0-412-28660-2.
^Troffaes, Matthias C. M.; Gert, de Cooman (2014). Lower previsions. ISBN 9780470723777.
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Global Information
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