Credal Set

A credal set is a set of probability distributions or, more generally, a set of (possibly 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.Cozman, F. (1999)
Theory of Sets of Probabilities (and related models) in a Nutshell
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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 $\mathrm$(X)/math>. In that case, the expectation for a function $f$ of $X$ with respect to the credal set $K(X)$ forms a closed interval underline[f\overline[f">.html" ;"title="underline[f">underline[f\overline[f, whose lower bound is called the lower prevision of $f$, and whose upper bound is called the upper prevision of $f$:
:\underline \min_ \int f \, d\mu=\min_ \int f \, d\mu
where $\mu$ denotes a probability measure, and with a similar expression for \overline /math> (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 \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 $\mathbb$ 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

References

Further reading

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Bayesian inference
Probability bounds analysis
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