## Generalized Bayesian inference in case of fuzzy information

The generalization of Bayes' theorem by application of the extension principle from fuzzy set theory is not reasonable, because it is not keeping the sequential updating procedure of Bayesian inference. Therefore another method was developed which takes care of imprecision of a-priori distributions and fuzziness of data as well. This is presented in the paper [VH04].

It is important to note that a more general concept of probability distributions, so-called fuzzy probability distributions, is necessary to describe imprecision of a-priori distributions.

A fuzzy probability distribution P*, defined on a sigma field A of subsets of an observation space M, is defined in the following way:

Definition 2. A fuzzy number is defined by a specialized characterizing function from section 2, for which all S-cuts are non-empty compact intervals.

Definition 3. A fuzzy probability distribution P* assigns to every event A gA a fuzzy number whose support is a subset of [0, 1] and which obeys:

1. P*(0) =0 and P*(M) = 1, i.e. the extreme events have precise probabilities

2. For any sequence A\,A2,- • • of pairwise disjoint events from A, and for all S-cuts Cg to be valid:

3. Fuzzy monotony: For A Ç B it follows P*(A) is fuzzy smaller than P *(B), i. e. forth e ô-cuts Cs [P*(A)] = [Ps (A),Ps (A)] and Cs [P*(B)] = [P_g(B),Pg(B)] the following has to be fulfilled:

Ps (A) < Ps (B) and Ps (A) < Ps (B) V ô £ (0,1] More details can be found in the paper [TH04].

Remark ,5. Special fuzzy probability distributions are obtained by so-called fuzzy density functions, which are also explained in [TH04].

### 6 Conclusion

Especially for quality of life data non-precise numbers are a more realistic description of quantitative data than precise real numbers. Generalized statistical analysis methods for this kind of data are available and provide valuable information in order to support well founded decisions.

### References

[FV04] Filzmoser, P., Viertl, R.: Testing hypotheses with fuzzy data: The fuzzy p-value. Metrika, 59 (2004) [OK95] Onisawa, T., Kacprzyk, J. (Eds.): Reliability and Safety Analyses under Fuzziness. Physica-Verlag, Heidelberg (1995) [TH04] Trutschnig, W., Hareter, D.: Fuzzy Probability Distributions. In: Lopéz-Diaz, M. et al. (eds.) Soft Methodology and Random Information Systems. Springer-Verlag, Berlin (2004) [Vie96] Viertl, R.: Statistical Methods for Non-Precise Data. CRC Press,

Boca Raton, Florida (1996) [Vie99] Viertl, R.: Nonprecise Data. in: Encyclopedia of Statistical Sciences

- Update, Volume 3. Wiley, New York (1999) [Vie02] Viertl, R.: On the Description and Analysis of Measurements of

Continuous Quantities. Kybernetika, 38 (2002) [VH04] Viertl, R., Hareter, D.: Fuzzy information and imprecise probability. ZAMM-Journal of Applied Mathematics and Mechanics, 84, No. 1011 (2004)