O'Reilly logo

Statistical Methods for Fuzzy Data by Reinhard Viertl

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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 (·; *) and δ(·; *) for all δ ∈ (0; 1].

For the δ-cuts of the fuzzy value l*(θ; *) we have

Unnumbered Display Equation

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.

Remark 15.1

The generalized likelihood function l*(θ; *) is a fuzzy valued function in the sense of Section 3.6, ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required