18.2 Fuzzy utility
In reality it is frequently not possible to find exact utility values. Therefore a more realistic approach to describe utilities is to use fuzzy utility values U*(θ, d) which are fuzzy intervals. Moreover, in general the uncertainty concerning the state of the system can be fuzzy too. This can be expressed by a fuzzy probability distribution P* on Θ, for example a fuzzy a posteriori distribution π*(·|D*).
In this situation, in order to find Bayesian decisions generalized expected utilities have to be calculated, i.e.
For classical probability distributions P on Θ the generalized expected utility can be calculated using the generalized integral from Section 3.6:
is calculated using the δ-level functions
The calculation of the above generalized Stieltjes integral is explained separately for discrete Θ and continuous Θ in the next two sections.