Empirical Similarity

An agent is asked to assess a real-valued variable Yp based on certain characteristics Xp=(Xp1,…,Xpm), and on a database consisting of (Xi1,…,Xim,Yi) for i = 1, …, n. A possible approach to combine past observations of X and Y with the current values of X to generate an assessment of Y is similarity-weighted averaging. It suggests that the predicted value of Y, Y¯ps, be the weighted average of all previously observed values Yi, where the weight of Yi, for every i = 1, …, n, is the similarity between the vector Xp1,…,Xpm, associated with Yp, and the previously observed vector, Xi1,…,Xim. We axiomatize this rule. We assume that, given every database, a predictor has a ranking over possible values, and we show that certain reasonable conditions on these rankings imply that they are determined by the proximity to a similarity-weighted average for a certain similarity function. The axiomatization does not suggest a particular similarity function, or even a particular form of this function. We therefore proceed to suggest that the similarity function be estimated from past observations. We develop tools of statistical inference for parametric estimation of the similarity function, for the case of a continuous as well as a discrete variable. Finally, we discuss the relationship of the proposed method to other methods of estimation and prediction.