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 ${\displaystyle \mathrm {ext} [K(X)]}$. 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 bound,

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

Notably, such an 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
• probability box
• robust Bayes analysis
• 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.

Further reading

• Abellán, J. N.; Moral, S. N. (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.