Axiomatization of an Exponential Similarity Function

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Discussion Paper

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An agent is asked to assess a real-valued variable y based on certain characteristics x = ( x 1 ,…, x m ), and on a database consisting of n observations of ( x 1 ,…, x m ,y). 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 s n +1, be the weighted average of all previously observed values y i , where the weight of y i is the similarity between the vector x 1 n +1 ,…, x m n +1, associated with y n +1, and the previously observed vector, x 1 i ,…, x m i . This paper axiomatizes, in terms of the prediction y n +1, a similarity function that is a (decreasing) exponential in a norm of the difference between the two vectors compared.

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