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Statistical Methods for Fuzzy Data by Reinhard Viertl

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22.3 Fuzzy predictive distributions

Predictive distributions for the dependent variable y in a Bayesian regression model are described in Section 21.4.

In the case of fuzzy data the a posteriori distribution is a fuzzy density π*(θ|z*).

The predictive density px(·|z) from Section 21.4 has to be generalized to the situation of fuzzy data z*. Therefore the definition of the predictive density from Section 21.4, i.e.

Unnumbered Display Equation

has to be adapted to the case of fuzzy data z*.

This is possible using the integration of fuzzy valued functions described in Section 3.6. The defining equation for the predictive density becomes in the case of fuzzy a posterior density π*(·|z*):

Unnumbered Display Equation

This generalized integration is conducted by the integration of δ-level functions, and then applying the construction lemma for characterizing functions.

The defining nested family of subsets of for with is obtained by the following equations:

From this the characterizing function ρ(·) of px*(y|z*) is constructed by

The δ-level ...

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