Optimal categorization

This paper studies categorizations that are optimal for the purpose of making predictions. A subject encounters an object (x,y). She observes the first component, x, and has to predict the second component, y. The space of objects is partitioned into categories. The subject determines what category the new object belongs to on the basis of x, and predicts that its y-value will be equal to the average y-value among the past observations in that category. The optimal categorization minimizes the expected prediction error. The main results are driven by a bias-variance trade-off: The optimal size of a category around x, is increasing in the variance of y conditional on x, decreasing in the variance of the conditional mean, decreasing in the size of the data base, and decreasing in the marginal density over x.