In mathematics, a credal set is a set of probability distributions 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. If a credal set K ( X ) {\displaystyle K(X)} is closed and convex, then, by the Krein–Milman theorem, it can be equivalently described by its extreme points e x t [ K ( X ) ] {\displaystyle \mathrm {ext} [K(X)]} . In that case, the expectation for a function f {\displaystyle f} of X {\displaystyle X} with respect to the credal set K ( X ) {\displaystyle K(X)} forms a closed interval [ E _ [ f ] , E ¯ [ f ] ] {\displaystyle [{\underline {E}}[f],{\overline {E}}[f]]} , whose lower bound is called the lower prevision of f {\displaystyle f} , and whose upper bound is called the upper prevision of f {\displaystyle f} :
E _ [ f ] = min μ ∈ K ( X ) ∫ f d μ = min μ ∈ e x t [ K ( X ) ] ∫ f d μ {\displaystyle {\underline {E}}[f]=\min _{\mu \in K(X)}\int f\,d\mu =\min _{\mu \in \mathrm {ext} [K(X)]}\int f\,d\mu }
where μ {\displaystyle \mu } denotes a probability measure, and with a similar expression for E ¯ [ f ] {\displaystyle {\overline {E}}[f]} (just replace min {\displaystyle \min } by max {\displaystyle \max } in the above expression). If X {\displaystyle X} is a categorical variable, then the credal set K ( X ) {\displaystyle K(X)} can be considered as a set of probability mass functions over X {\displaystyle X} . If additionally K ( X ) {\displaystyle K(X)} is also closed and convex, then the lower prevision of a function f {\displaystyle f} of X {\displaystyle X} can be simply evaluated as:
E _ [ f ] = min p ∈ e x t [ K ( X ) ] ∑ x f ( x ) p ( x ) {\displaystyle {\underline {E}}[f]=\min _{p\in \mathrm {ext} [K(X)]}\sum _{x}f(x)p(x)}
where p {\displaystyle p} denotes a probability mass function. It is easy to see that a credal set over a Boolean variable X {\displaystyle X} cannot have more than two extreme points (because the only closed convex sets in R {\displaystyle \mathbb {R} } are closed intervals), while credal sets over variables X {\displaystyle 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 Abellán, Joaquín; Moral, Serafín (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.