Credal set

A credal set is a set of probability distributions^[1] or, equivalently, a set of 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]

Let X denote a categorical variable, P(X) a probability mass function over X, and K(X) a credal set over X. If K(X) is convex, the credal set can be equivalently described by its extreme points ext[K(X)]. The expectation for a function f of X with respect to the credal set K(X) can be characterised only by its lower and upper bounds. For the lower we have $\underline{E}[f]=\min_{P(X)\in K(X)} \sum_x f(x) P(x)$. Notably, such a inference problem can be equivalently obtained by considering only the extreme points of the credal set.

It is easy to see that a credal set over a Boolean variable cannot have more than two vertices, while no bounds can be provided for credal sets over variables with three or more values.

See also

• imprecise probability
• Dempster–Shafer theory
• p-boxes
• robust Bayes
• upper and lower probabilities

References

1. ^ Levi, I. (1980). The Enterprise of Knowledge. MIT Press, Cambridge, Massachusetts.
2. ^ Cozman, F. (1999). Theory of Sets of Probabilities (and related models) in a Nutshell.

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