Credal set

In mathematics, a credal set is a set of probability distributions 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.

If a credal set $$K(X)$$ is closed and convex, then, by the Krein–Milman theorem, it can be equivalently described by its extreme points $$\mathrm{ext}[K(X)]$$. In that case, the expectation for a function $$f$$ of $$X$$ with respect to the credal set $$K(X)$$ forms a closed interval $$[\underline{E}[f],\overline{E}[f]]$$, whose lower bound is called the lower prevision of $$f$$, and whose upper bound is called the upper prevision of $$f$$:
 * $$\underline{E}[f]=\min_{\mu\in K(X)} \int f \, d\mu=\min_{\mu\in \mathrm{ext}[K(X)]} \int f \, d\mu$$

where $$\mu$$ denotes a probability measure, and with a similar expression for $$\overline{E}[f]$$ (just replace $$\min$$ by $$\max$$ in the above expression).

If $$X$$ is a categorical variable, then the credal set $$K(X)$$ can be considered as a set of probability mass functions over $$X$$. If additionally $$K(X)$$ is also closed and convex, then the lower prevision of a function $$f$$ of $$X$$ can be simply evaluated as:
 * $$\underline{E}[f]=\min_{p\in \mathrm{ext}[K(X)]} \sum_x f(x) p(x)$$

where $$p$$ denotes a probability mass function. It is easy to see that a credal set over a Boolean variable $$X$$ cannot have more than two extreme points (because the only closed convex sets in $$\mathbb{R}$$ are closed intervals), while credal sets over variables $$X$$ that can take three or more values can have any arbitrary number of extreme points.