Coarse-grained information in formal theory of measurement

Measurement procedure is not determined only by measured objects, but is subject to various factors such as adopted instruments, and hence, the result of measurement reflects, as well as quantitative aspect of measured objects, the influence of those additional factors, one of which is the degree of the fineness of observation. The present paper is an effort to introduce this subject into a formal framework of measurement science. First, the nature of information obtained by nominal scale is examined, and in what sense the nominal scale is informative is clarified. Then, information obtained by coarse nominal measurement, where some symbols are put into one group as the same symbol, is discussed, and a possible approach to group-formation procedure as a model of coarse observation of objects is proposed. Information-theoretical interdependence is often measured by using entropy function. However, the results suggest that the average entropy is rather a suitable measure of the degree of interdependence.