Axiomatization of an exponential similarity function

An individual is asked to assess a real-valued variable y based on certain characteristics x=(x1 ,..., xm ), and on a database consisting of n observations of (x1 ,..., xm , 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, yn+1 s , be the weighted average of all previously observed values yi, where the weight of yi is the similarity between the vector xn+1 1 ,..., xn+1 m , associated with yn+1, and the previously observed vector, xi 1 ,..., xi m . This paper axiomatizes, in terms of the prediction yn+1, a similarity function that is a (decreasing) exponential in a norm of the difference between the two vectors compared.

• Type: Journal
• Year: 2008
• Where: MSS
• Authors: Antoine Billot, Itzhak Gilboa, David Schmeidler
• Keywords: Certain Characteristics, Hardware, MSS 2008, Observations, Possible Approach