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 $K(X)$ 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 $f$ :^[3]

{\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 $X$ .^[4] 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.^{[citation needed]}

References

^ Levi, Isaac (1980). The Enterprise of Knowledge. MIT Press, Cambridge, Massachusetts.
^ Cozman, Fabio (1999). Theory of Sets of Probabilities (and related models) in a Nutshell Archived 2011-07-21 at the Wayback Machine.
^ Walley, Peter (1991). Statistical Reasoning with Imprecise Probabilities. London: Chapman and Hall. ISBN 0-412-28660-2.
^ Troffaes, Matthias C. M.; de Cooman, Gert (2014). Lower previsions. ISBN 9780470723777.

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