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.[citation needed]

See also

Further reading