Fuzziness and loss of information in statistical problems

Statistical problems in which the available experimental observations are not necessarily exact have been considered previously. These problems have been described by means of a probabilistic-possibilistic approach based on the concept of the “fuzzy information system.” When Zadeh's probabilistic definition is adopted, the gain of information about the state of nature or parameter contained in a fuzzy information system and the uncertainty associated with an inference from it could be quantified by means of several concepts: the quantity of information, the worth of information, the set of attainable loss vectors, the expected quietness of information, the risk function, the mean-squared error, and the probabilities of type I and type II errors. The author proves that the fuzziness in the experimental observations entails a loss of information about the state or parameter space, whatever the quantification of the gain of information or the uncertainty may be.