15.2 Bayes’ theorem for fuzzy a priori distribution and fuzzy data

Using the averaging procedure of δ-level curves of the a priori density from Section 14.2 and combining it with the generalized likelihood function from Section 15.1, the generalization of Bayes’ theorem is possible. The construction is based on δ-level functions.

Based on a fuzzy a priori density π*(·) on Θ with δ-level functions δ(·) and δ(·), and a fuzzy sample x1*, …, xn* with combined fuzzy sample * whose vector-characterizing function is ξ(., …, .), the characterizing function ψl*(θ; *)(·) of l*(θ; *) is obtained by the extension principle from Section 3.1, i.e.

Unnumbered Display Equation

The δ-level curves of the fuzzy a posteriori density π*(·|x1*, … xn*) = π*(·|*) are defined in the following way:

and

for all δ ∈ (0; 1].

Remark 15.2

For the ...

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