Credal set explained
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
is closed and convex, then, by the
Krein–Milman theorem, it can be equivalently described by its
extreme points
. In that case, the expectation for a function
of
with respect to the credal set
forms a closed interval
[\underline{E}[f],\overline{E}[f]]
, whose lower bound is called the lower prevision of
, and whose upper bound is called the upper prevision of
:
[3] \underline{E}[f]=min\mu\in\intfd\mu=min\mu\in\intfd\mu
where
denotes a
probability measure, and with a similar expression for
(just replace
by
in the above expression).
If
is a
categorical variable, then the credal set
can be considered as a set of
probability mass functions over
.
[4] If additionally
is also closed and convex, then the lower prevision of a function
of
can be simply evaluated as:
\underline{E}[f]=minp\in\sumxf(x)p(x)
where
denotes a
probability mass function.It is easy to see that a credal set over a
Boolean variable
cannot have more than two extreme points (because the only closed convex sets in
are closed intervals), while credal sets over variables
that can take three or more values can have any arbitrary number of extreme points.
See also
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 .
- Book: Walley
, Peter
. Statistical Reasoning with Imprecise Probabilities. Chapman and Hall. 1991. London. 0-412-28660-2 .
- Book: Troffaes. Matthias C. M.. Gert. de Cooman. Lower previsions. 2014. 9780470723777 .
Further reading
