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 [math]\displaystyle{ K(X) }[/math] is closed and convex, then, by the Krein–Milman theorem, it can be equivalently described by its extreme points [math]\displaystyle{ \mathrm{ext}[K(X)] }[/math]. In that case, the expectation for a function [math]\displaystyle{ f }[/math] of [math]\displaystyle{ X }[/math] with respect to the credal set [math]\displaystyle{ K(X) }[/math] forms a closed interval [math]\displaystyle{ [\underline{E}[f],\overline{E}[f]] }[/math], whose lower bound is called the lower prevision of [math]\displaystyle{ f }[/math], and whose upper bound is called the upper prevision of [math]\displaystyle{ f }[/math]:[3]
- [math]\displaystyle{ \underline{E}[f]=\min_{\mu\in K(X)} \int f \, d\mu=\min_{\mu\in \mathrm{ext}[K(X)]} \int f \, d\mu }[/math]
where [math]\displaystyle{ \mu }[/math] denotes a probability measure, and with a similar expression for [math]\displaystyle{ \overline{E}[f] }[/math] (just replace [math]\displaystyle{ \min }[/math] by [math]\displaystyle{ \max }[/math] in the above expression).
If [math]\displaystyle{ X }[/math] is a categorical variable, then the credal set [math]\displaystyle{ K(X) }[/math] can be considered as a set of probability mass functions over [math]\displaystyle{ X }[/math].[4] If additionally [math]\displaystyle{ K(X) }[/math] is also closed and convex, then the lower prevision of a function [math]\displaystyle{ f }[/math] of [math]\displaystyle{ X }[/math] can be simply evaluated as:
- [math]\displaystyle{ \underline{E}[f]=\min_{p\in \mathrm{ext}[K(X)]} \sum_x f(x) p(x) }[/math]
where [math]\displaystyle{ p }[/math] denotes a probability mass function. It is easy to see that a credal set over a Boolean variable [math]\displaystyle{ X }[/math] cannot have more than two extreme points (because the only closed convex sets in [math]\displaystyle{ \mathbb{R} }[/math] are closed intervals), while credal sets over variables [math]\displaystyle{ X }[/math] that can take three or more values can have any arbitrary number of extreme points.[citation needed]
See also
- Imprecise probability
- Dempster–Shafer theory
- Probability box
- Robust Bayes analysis
- Upper and lower probabilities
References
- ↑ Levi, I. (1980). The Enterprise of Knowledge. MIT Press, Cambridge, Massachusetts.
- ↑ Cozman, F. (1999). Theory of Sets of Probabilities (and related models) in a Nutshell .
- ↑ Walley, Peter (1991). Statistical Reasoning with Imprecise Probabilities. London: Chapman and Hall. ISBN 0-412-28660-2.
- ↑ Troffaes, Matthias C. M.; Gert, de Cooman (2014). Lower previsions. ISBN 9780470723777.
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.
 |  |
