Axiomatization of an Exponential Similarity Function

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_{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+1}^{1},...,x_{n+1}^{m}, associated with y_{n+1}, and the previously observed vector, x_{i}^{1},...,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.