14
Bayes’ theorem and fuzzy information
Looking at real data and realistic a priori distributions two kinds of fuzzy information appear. First, fuzzy samples and secondly, fuzzy a priori distributions. The necessity of generalizing Bayesian inference was pointed out in Viertl (1987).
The description of fuzzy data is given in earlier parts of this book. Fuzzy a priori information can be expressed by fuzzy probability distributions in the sense of Chapter .
14.1 Fuzzy a priori distributions
In the case of discrete stochastic model p(.|θ); θ ∈ Θ and discrete parameter space Θ = {θ1, … , θk} fuzzy a priori information concerning the parameter can be expressed by k fuzzy intervals π*(θ1), … , π*(θk) with π*(θj) = Pr{θj} for which π* (θ1) ⊕ … ⊕ π*(θk) is a fuzzy interval whose characterizing function η (·) fulfills 1 ∈ C1 [η(·)] and the characterizing functions ξj (·) of π*(θj) are fulfilling the following:
For each j ∈ {1, … , k} there exists a number
such that
Since Cδ [π*(θj)] are closed intervals for all δ ∈ (0; 1], the fuzzy probability of a subset Θ1 ⊂ Θ with Θ1 = {θj1, … , θjm} is denoted by P*(Θ1). The δ-cuts of P*(Θ1) are defined to be the set of all possible sums of numbers xj ∈ Cδ(pjl*), l = 1(1)m, obeying
Remark 14.1:
From the above definition it follows that P*() = 0 and ...