Bayesian regression analysis and fuzzy information
In Bayesian regression analysis two kinds of fuzziness are present: Fuzziness of a priori distributions and fuzziness of data. In this chapter we assume that fuzzy a priori distributions are given by fuzzy densities π*(·) with corresponding δ-level functions δ(·) and δ(·), δ ∈ (0;1). The data are given in the form of n vectors of fuzzy numbers, i.e.
or in the form of fuzzy vectors for the independent variable x = (x1,…,xk), and fuzzy number yi* for the dependent variable y:
where xi* is a k-dimensional fuzzy vector.
In order to apply Bayes’ theorem first the likelihood function has to be generalized for fuzzy data. In accordance with Chapter the combined fuzzy sample has to be formed.
In the first situation of data (xi*1,…,xi*k, yi*), i = 1(1)n the combined fuzzy sample z* is an n × (k + 1 )-dimensional fuzzy vector whose vector-characterizing function ζ(x11,…,xnk, Y1,…,Yn) is formed by the minimum t-norm, i.e.
where ξij(·) are the ...