O'Reilly logo

Statistical Methods for Fuzzy Data by Reinhard Viertl

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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

Unnumbered Display Equation

such that

Unnumbered Display Equation

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*1). The δ-cuts of P*1) are defined to be the set of all possible sums of numbers xjCδ(pjl*), l = 1(1)m, obeying

Unnumbered Display Equation

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.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required