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 P*(Θ) = 1, and the inequalities from Remark 8.2 in Chapter 8.

For continuous parameter space Θ a fuzzy a priori distribution P*(·) on Θ is given by a fuzzy valued density function π*(·) on Θ as described in Section 8.1.

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