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 $\mathrm{e}\mathrm{x}\mathrm{t}[K(X)]$. In that case, the expectation for a function $f$ of $X$ with respect to the credal set $K(X)$ forms a closed interval $[\underset{\_}{E}[f],\bar{E}[f]]$, whose lower bound is called the lower prevision of $f$, and whose upper bound is called the upper prevision of $f$:

$\underset{\_}{E}[f]=\underset{\mu \in K(X)}{\min }\int fd\mu =\underset{\mu \in \mathrm{e}\mathrm{x}\mathrm{t}[K(X)]}{\min }\int fd\mu$

where $\mu$ denotes a probability measure, and with a similar expression for $\bar{E}[f]$ (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:

$\underset{\_}{E}[f]=\underset{p\in \mathrm{e}\mathrm{x}\mathrm{t}[K(X)]}{\min }\sum _{x}f(x)p(x)$

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.

See also

• Imprecise probability
• Dempster–Shafer theory
• Probability box
• Robust Bayes analysis
• Upper and lower probabilities

Further reading

• Abellán, Joaquín; Moral, Serafín (2005). "Upper entropy of credal sets. Applications to credal classification". International Journal of Approximate Reasoning. 39 (2–3): 235. doi:10.1016/j.ijar.2004.10.001.