Credal set
# 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.

