Quantity of information; comparison between information systems. II. Fuzzy states. (English) Zbl 0559.94003

Gil, M. A.; Lopez, M. T.; Gil, P.

Fuzzy Sets Syst. 15, 129-145 (1985).

One of the basic purposes in statistical theory is to get information about the true state of nature. In order to obtain this information we can usually perform certain experiments for which the distribution depends upon the true state and, consequently, they reduce the amount of uncertainty associated with the set of possible states (state space). When the information provided by the performance of a potential experiment can only be known in an approximate way, we assimilate this information with a fuzzy restriction or fuzzy information [H. Tanaka, T. Okuda and K. Asai, Adv. in fuzzy set theory and appl., 303-320 (1979; Zbl 0434.94026)], and the set of all possible informations with a fuzzy information system (f.i.s.). In this paper, we consider the problem of obtaining information about the original state space (nonfuzzy state space, Part I) resp. about certain vague states (fuzzy states, Part II) in the situation we have just described. We then suggest a selection among the available experiments providing fuzzy information on the basis of the quantity of information of a f.i.s. concerning the state space (as defined by H. Tanaka et al. [loc. cit.]). This selection extends the preference relation stated by D. V. Lindley [Ann. Math. Stat. 27, 986-1005 (1956; Zbl 0073.141)] to the fuzzy framework.

MSC:

94A17	Measures of information, entropy
62B10	Statistical aspects of information-theoretic topics
94D05	Fuzzy sets and logic (in connection with information, communication, or circuits theory)
62B15	Theory of statistical experiments

Keywords:

experiments; amount of uncertainty; state space; fuzzy information system; quantity of information; preference relation

Citations:

Zbl 0559.94002; Zbl 0434.94026; Zbl 0073.141

References:

[1]	Blackwell, D., Comparisons of experiments, (), 93-102
[2]	Bouchon, B.; Cohen, G., Partitions and fuzziness, (1981), submitted
[3]	Bouchon, B., Comparison of experiments and improvement of models, (), 172-175
[4]	García-Carrasco, M.P., Criterios para la comparación de experimentos, (), 28-51
[5]	Gil, M.A., Criterion of maximizing the expected quietness (invariant by homotethies in relation to the utilities, R.A.I.R.O. rech oper., 16, 319-331, (1982) · Zbl 0505.62002
[6]	Gil, M.A., Mixed criterion of expected utility and quietness (invariant by homotethies with respect to the utilities, Statistica (Bologna), 42, 1, 21-37, (1982) · Zbl 0504.62009
[7]	Gil, P., Medidas de incertidumbre e información en problemas de decísion estadística, Rev. acad. ciencias Madrid, 69, 3, 549-610, (1975)
[8]	Giles, R., Lukasiewicz logic and fuzzy theory, Internat. J. man. Mach. stud., 8, 313-327, (1976) · Zbl 0335.02037
[9]	Goguen, J.A., \(L\)-fuzzy sets, J. math. anal. appl., 18, 145-174, (1967) · Zbl 0145.24404
[10]	Hisdal, E., Possibilities and probabilities, (), 341-345
[11]	De Luca, A.; Termini, S., A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory, Inform. control., 20, 301-312, (1972) · Zbl 0239.94028
[12]	Okuda, T.; Tanaka, H.; Asai, K., A formulation of fuzzy decision problems with fuzzy information, using probability measures of fuzzy events, Inform. control, 38, 135-147, (1978) · Zbl 0401.94050
[13]	Tanaka, H.; Okuda, T.; Asai, K., A formulation of fuzzy decision problems and its applications to an investment problems, Kybernetes, 5, 30-52, (1976) · Zbl 0332.90001
[14]	Tanaka, H.; Okuda, T.; Asai, K., Fuzzy information and decision in statistical model, (), 303-320
[15]	Warren, R.H., Equivalent fuzzy sets (short communication, Fuzzy sets and systems, 6, 309-312, (1981) · Zbl 0464.04005
[16]	Zadeh, L.A., Fuzzy sets, Inform. control, 8, 338-353, (1965) · Zbl 0139.24606
[17]	Zadeh, L.A., Probability measures of fuzzy events, J. math. anal. appl., 23, 421-427, (1968) · Zbl 0174.49002
[18]	Zadeh, L.A., Theory of fuzzy sets, () · Zbl 0377.04002
[19]	Gil, M.A.; López, M.T.; Gil, P., Quantity of information; comparison between information systems: I. non-fuzzy states, Fuzzy sets and systems, 15, 65-78, (1985) · Zbl 0559.94002