Credal set

In mathematics, a credal set is a set of probability distributionsLevi, Isaac (1980). The Enterprise of Knowledge. MIT Press, Cambridge, Massachusetts. 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.Cozman, Fabio (1999). [http://www.poli.usp.br/p/fabio.cozman/Research/CredalSetsTutorial/Nutshell/index.html Theory of Sets of Probabilities (and related models) in a Nutshell] {{webarchive|url=https://web.archive.org/web/20110721192633/http://www.poli.usp.br/p/fabio.cozman/Research/CredalSetsTutorial/Nutshell/index.html |date=2011-07-21 }}.

If a credal set $K(X)$ is closed and convex, then, by the Krein–Milman theorem, it can be equivalently described by its extreme points $\mathrm{ext}[K(X)]$ . In that case, the expectation for a function $f$ of $X$ with respect to the credal set $K(X)$ forms a closed interval $[\underline{E}[f],\overline{E}[f]]$ , whose lower bound is called the lower prevision of $f$ , and whose upper bound is called the upper prevision of $f$ :{{cite book

| last = Walley

| first = Peter

| title = Statistical Reasoning with Imprecise Probabilities

| publisher = Chapman and Hall

| year = 1991

| location = London

| isbn = 0-412-28660-2 }}

: $\underline{E}[f]=\min_{\mu\in K(X)} \int f \, d\mu=\min_{\mu\in \mathrm{ext}[K(X)]} \int f \, d\mu$

where $\mu$ denotes a probability measure, and with a similar expression for $\overline{E}[f]$ (just replace $\min$ by $\max$ in the above expression).

If $X$ is a categorical variable, then the credal set $K(X)$ can be considered as a set of probability mass functions over $X$ .{{cite book

| last1 = Troffaes

| first1 = Matthias C. M.

| first2 = Gert

| last2 = de Cooman

| title = Lower previsions

| year = 2014

| isbn = 9780470723777 }}

If additionally $K(X)$ is also closed and convex, then the lower prevision of a function $f$ of $X$ can be simply evaluated as:

: $\underline{E}[f]=\min_{p\in \mathrm{ext}[K(X)]} \sum_x f(x) p(x)$

where $p$ denotes a probability mass function.

It is easy to see that a credal set over a Boolean variable $X$ cannot have more than two extreme points (because the only closed convex sets in $\mathbb{R}$ are closed intervals), while credal sets over variables $X$ that can take three or more values can have any arbitrary number of extreme points.{{cn|date=January 2012}}

Credal set

See also

References

Further reading