Empirical Similarity

Abstract

International audienceAn agent is asked to assess a real-valued variable Yp based on certain characteristics Xp = (Xp-super-1, ..., Xp-super-m), and on a database consisting of Xi-super-1, ... Xi-super-m, 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, Ȳp-super-s, 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 Xp-super-1, ..., Xp-super-m, associated with Yp, and the previously observed vector, Xi-super-1, ..., Xi-super-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. Copyright by the President and Fellows of Harvard College and the Massachusetts Institute of Technology

Probabilities as Similarity-Weighted Frequencies

“…We mention it here to highlight the fact that the case n = 2 is not covered by our result. See Gilboa, Lieberman, and Schmeidler (2004).…”

“…Hence, the similarity function should be estimated from past data. Gilboa, Lieberman, and Schmeidler (2004) axiomatize formula (1) for the case n = 2 (not dealt with in this paper) and develop the statistical theory required for the estimation of the function s, assuming that such a function governs the data generating process. The present paper provides an axiomatization for the case n > 2.…”

Imprecise Data Sets as a Source of Ambiguity: A Model and Experimental Evidence

“…First, in the absence of a theory of belief formation, one cannot tell which beliefs are reasonable in a given context. This point was forcefully made by Gilboa, Postlewaite, and Schmeidler (2004), and it seems to be a major motivation for recent derivations of case-based probabilities by Billot, Gilboa, Samet, and Schmeidler (2005) and Gilboa, Lieberman, and Schmeidler (2006). We maintain that a theory of belief formation which restricts the range of reasonable beliefs is just as important when discussing non-Bayesian beliefs.…”

A Multivariate Kernel Approach to Forecasting the Variance Covariance of Stock Market Returns

“…This principle is extended to the multivariate setting in this paper with a kernel based approach proposed for forecasting the VCM and is an application of the general technique of empirical similarity, as described more generally in Gilboa et al (2006). The kernel density acts as a similarity function and the forecasts of the rVCM are similarity weighted averages.…”