Empirical Similarity

An agent is asked to assess a real-valued variable Y_p based on certain characteristics X_p = (X_p¹, …, X_p^m), and on a database consisting of X_i¹, … X_i^m, Y_i) 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, Ȳ_p^s, be the weighted average of all previously observed values Y_i, where the weight of Y_i for every i = 1, …, n, is the similarity between the vector X_p¹, …, X_p^m, associated with Y_p, and the previously observed vector, X_i¹, …, X_i^m. 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.