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 procedure in Bayes’ theorem, the generalization of Bayes’ theorem to the situation of fuzzy a priori distribution and fuzzy data is possible.
The generalized likelihood function l*(θ; *) is a fuzzy valued function in the sense of Section 3.6, ...