Credal set
A credal set is a set of probability distributions or, more generally, a set of 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.
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 :
where denotes a probability measure, and with a similar expression for .
If is a categorical variable, then the credal set can be considered as a set of probability mass functions over.
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, while credal sets over variables that can take three or more values can have any arbitrary number of extreme points.
