Axiomatization of an exponential similarity function

✍ Scribed by Antoine Billot; Itzhak Gilboa; David Schmeidler

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 165 KB
• Volume: 55
• Category: Article
• ISSN: 0165-4896
• DOI: 10.1016/j.mathsocsci.2007.08.002

✦ Synopsis

An individual 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.