An agent is asked to assess a real-valued variable Y p based on certain characteristics X p = (X p-super-1, ..., X p-super-m), and on a database consisting of X i-super-1, ... X i-super-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, È² p-super-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 p-super-1, ..., X p-super-m, associated with Y p, and the previously observed vector, X i-super-1, ..., X i-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.
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- Billot, Antoine & Gilboa, Itzhak & Schmeidler, David, 2008.
"Axiomatization of an exponential similarity function,"
Mathematical Social Sciences,
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- Antoine Billot & Itzhak Gilboa & David Schmeidler, 2004. "Axiomatization of an Exponential Similarity Function," Cowles Foundation Discussion Papers 1485, Cowles Foundation for Research in Economics, Yale University.
- Gabrielle Gayer & Itzhak Gilboa & Offer Lieberman, 2004.
"Rule-Based and Case-Based Reasoning in Housing Prices,"
Cowles Foundation Discussion Papers
1493, Cowles Foundation for Research in Economics, Yale University.
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- Gabrielle Gayer & Itzhak Gilboa & Offer Lieberman, 2004. "Rule-Based and Case-Based Reasoning in Housing Prices," Levine's Bibliography 122247000000000672, UCLA Department of Economics.
- Antoine Billot & Itzhak Gilboa & David Schmeidler, 2004. "An Axiomatization of an Exponential Similarity Function," Levine's Bibliography 122247000000000678, UCLA Department of Economics.
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- Antoine Billot & Itzhak Gilboa & Dov Samet & David Schmeidler, 2003. "Probabilities: Frequencies Viewed in Perspective," Levine's Bibliography 666156000000000295, UCLA Department of Economics.
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