Probability-theoretic Definition of Category Utility
The probability-theoretic definition of category utility given in Fisher (1987) and Witten & Frank (2005) is as follows:
where is a size- set of -ary features, and is a set of categories. The term designates the marginal probability that feature takes on value, and the term designates the category-conditional probability that feature takes on value given that the object in question belongs to category .
The motivation and development of this expression for category utility, and the role of the multiplicand as a crude overfitting control, is given in the above sources. Loosely (Fisher 1987), the term is the expected number of attribute values that can be correctly guessed by an observer using a probability-matching strategy together with knowledge of the category labels, while is the expected number of attribute values that can be correctly guessed by an observer the same strategy but without any knowledge of the category labels. Their difference therefore reflects the relative advantage accruing to the observer by having knowledge of the category structure.
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