16.1 Bayesian confidence regions based on fuzzy data
In the case of classical a priori distributions the generalization to the situation of fuzzy samples x1*,…, xn* is possible using the combined fuzzy sample *, whose vector-characterizing is denoted by ξ(.,…,.). The generalization is similar to that in Section 12.2.
Let Θ,1−α be a standard Bayesian confidence region for θ based on a classical sample = (x1,…,xn).
The Bayesian confidence region for θ based on a fuzzy sample, whose combined fuzzy sample has vector-characterizing function ξ:MXn→[0;1], is a fuzzy subset Θ1−α* of Θ whose membership function φ(·) is given by its values φ(θ) in the following way:
For classical samples = (x1,…,xn) the membership function φ(·) reduces to the indicator function of Θ,1−α, i.e.
The proof ...