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A '''credal set''' is a set of [[probability distribution]]s<ref>Levi, I. (1980). ''The Enterprise of Knowledge''. MIT Press, Cambridge, Massachusetts. </ref> or, equivalently, a set of [[probability measure]]s. A credal set is often assumed or constructed to be a [[closure (mathematics)|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 probability|Bayesian]] agent about the possible states of the world.<ref>Cozman, F. (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].</ref>
 
Let <math>X</math> denote a [[categorical variable]], <math>P(X)</math> a [[probability mass function]] over <math>X</math>, and <math>K(X)</math> a credal set over <math>X</math>. If <math>K(X)</math> is convex, the credal set can be equivalently described by its extreme points <math>\mathrm{ext}[K(X)]</math>. The expectation for a function <math>f</math> of <math>X</math> with respect to the credal set <math>K(X)</math> can be characterised only by its lower and upper bounds. For the lower bound,  
:<math>\underline{E}[f]=\min_{P(X)\in K(X)} \sum_x f(x) P(x).</math>
Notably, such an inference problem can be equivalently obtained by considering only the extreme points of the credal set.{{cn|date=January 2012}}
 
It is easy to see that a credal set over a [[Boolean variable]] cannot have more than two vertices, while no bounds can be provided for credal sets over variables with three or more values.{{cn|date=January 2012}}
 
==See also==
* [[imprecise probability]]
* [[Dempster–Shafer theory]]
* [[probability box]]
* [[robust Bayes analysis]]
* [[upper and lower probabilities]]
 
==References==
<references/>
==Further reading==
*{{cite doi|10.1016/j.ijar.2004.10.001}}
 
[[Category:Bayesian inference]]
[[Category:Probability bounds analysis]]
{{probability-stub}}

Latest revision as of 03:13, 22 October 2013

A credal set is a set of probability distributions[1] or, equivalently, a set of 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]

Let X denote a categorical variable, P(X) a probability mass function over X, and K(X) a credal set over X. If K(X) is convex, the credal set can be equivalently described by its extreme points ext[K(X)]. The expectation for a function f of X with respect to the credal set K(X) can be characterised only by its lower and upper bounds. For the lower bound,

E_[f]=minP(X)K(X)xf(x)P(x).

Notably, such an inference problem can be equivalently obtained by considering only the extreme points of the credal set.Template:Cn

It is easy to see that a credal set over a Boolean variable cannot have more than two vertices, while no bounds can be provided for credal sets over variables with three or more values.Template:Cn

See also

References

  1. Levi, I. (1980). The Enterprise of Knowledge. MIT Press, Cambridge, Massachusetts.
  2. Cozman, F. (1999). Theory of Sets of Probabilities (and related models) in a Nutshell.

Further reading

Template:Probability-stub