Axiomatization of an exponential similarity function

An individual is asked to assess a real-valued variable y based on certain characteristics x = (x¹,..., x^m), and on a database consisting of n observations of (x¹,..., 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_n+1^s, be the weighted average of all previously observed values y_i, where the weight of y_i is the similarity between the vector x_n+1¹,..., x_n+1^m, associated with y_n+1, and the previously observed vector, x_i¹,..., x_i^m. 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.