Credal set

A credal set is a set of probability distributions or, equivalently, 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.

Let X denote a categorical variable, P ( X ) a probability mass function over X , and K ( X ) a credal set over X . If K ( X ) is convex, the credal set can be equivalently described by its extreme points e x t [ K ( X ) ] . The expectation for a function f of X with respect to the credal set K ( X ) can be characterised only by its lower and upper bounds. For the lower bound,

Notably, such an inference problem can be equivalently obtained by considering only the extreme points of the credal set.

It is easy to see that a credal set over a Boolean variable cannot have more than two vertices, while no bounds can be provided for credal sets over variables with three or more values.