The natural extension, the key concept for the construction of coherent imprecise models, can appear in different equivalent forms. Each of them has pros and cons in the context of specific applications. The use of a proper form can substantially facilitate the inference and computation of the previsions of interest. The current paper concerns four forms of the natural extension representation. It is demonstrated that all of them are equivalent and one, discussed in the last instance, is prominent solely for gambles defined on continuous possibility sets. It is proven that the solution of the natural extension problem for continuous gambles exists on the degenerate distributions. Partial information and a characteristic to be calculated can be thought as the expectations of some real-valued functions defined on the possibility space. As they are expectations, they can be expressed as functions of probability density functions and proper real-valued functions (gambles). Each piece of the partial information acts as a constraint to the probability distributions. All together the constraints define the area of all distributions over which the interval of the desired statistical characteristic will be searched. Throughout the paper the natural extension is analyzed through the prism of re-liability application.
Keywords. Imprecise probability theory, imprecise reliability, natural extension, previsions.
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Authors addresses:
Igor Kozine
Systems Analysis Department
Building 100
P.O. Box 49, 4000 Roskilde
Denmark
Lev Utkin
Department of Computer Science
Institutski per. 5
Saint Petersburg
194021, Russia
E-mail addresses:
| Igor Kozine | igor.kozine@risoe.dk |
| Lev Utkin | lvu@utkin.usr.etu.spb.ru |