15
Generalized Bayes’ theorem
For continuous stochastic models X ~ f(·|θ); θ ∈ Θ with continuous parameter space in general a priori distributions as well as observations are fuzzy. Therefore it is necessary to generalize Bayes’ theorem to this situation.
15.1 Likelihood function for fuzzy data
In the case of fuzzy data x1*, …, xn* the likelihood function l(θ;x1, …, xn) has to be generalized to the situation of fuzzy variables x1*, …, xn*. The basis for that is the combined fuzzy sample element 
* from Chapter . Then the generalized likelihood function l*(θ; *) is represented by its δ-level functions l-δ (·; *) and 
δ(·; *) for all δ ∈ (0; 1].
For the δ-cuts of the fuzzy value l*(θ; *) we have
Using this and the construction from Chapter in order to keep the sequential property of the updating ...
