18
Bayesian decisions and fuzzy information
Decisions are frequently connected with utility or loss. The quantification of the utility U(d) of a decision d can be a difficult task. Therefore a suitable model is to describe the utility by a fuzzy number U*(d). Moreover the utility of a decision depends on the state of the considered system. Therefore the utility depends on both the state Θ and the decision d, i.e. U(θ, d).
18.1 Bayesian decisions
In Bayesian decision analysis it is possible to include a priori information in the form of a priori distributions, and also a posteriori distributions on the space Θ of possible states of the considered decision situation.
Let Θ = Θ: Θ possible state be the state space, 
= {d : d possible decision} be the set of possible decisions, and P be a probability distribution on the state space Θ. Moreover let U(.,.) be a utility function, i.e.
where U(θ, d) denotes the utility of the decision d if the system is in state Θ.
Assuming that the corresponding sums or integrals exist, the basis for optimal decisions is the expected utility
Here 
, is the ...
