Credal set

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 ${\displaystyle K(X)}$ is closed and convex, then, by the Krein–Milman theorem, it can be equivalently described by its extreme points ${\displaystyle \mathrm {ext} [K(X)]}$. In that case, the expectation for a function ${\displaystyle f}$ of ${\displaystyle X}$ with respect to the credal set ${\displaystyle K(X)}$ forms a closed interval ${\displaystyle [{\underline {E}}[f],{\overline {E}}[f]]}$, whose lower bound is called the lower prevision of ${\displaystyle f}$, and whose upper bound is called the upper prevision of ${\displaystyle f}$:[3]

${\displaystyle {\underline {E}}[f]=\min _{\mu \in K(X)}\int f\,d\mu =\min _{\mu \in \mathrm {ext} [K(X)]}\int f\,d\mu }$

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

If ${\displaystyle X}$ is a categorical variable, then the credal set ${\displaystyle K(X)}$ can be considered as a set of probability mass functions over ${\displaystyle X}$.[4] If additionally ${\displaystyle K(X)}$ is also closed and convex, then the lower prevision of a function ${\displaystyle f}$ of ${\displaystyle X}$ can be simply evaluated as:

${\displaystyle {\underline {E}}[f]=\min _{p\in \mathrm {ext} [K(X)]}\sum _{x}f(x)p(x)}$

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