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

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.


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
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