Axiomatization of an Exponential Similarity Function

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