A decision maker is asked to express her beliefs by assigning probabilities to certain possible states. We focus on the relationship between her database and her beliefs. We show that, if beliefs given a union of two databases are a convex combination of beliefs given each of the databases, the belief formation process follows a simple formula: beliefs are a similarity-weighted average of the beliefs induced by each past case.
An agent 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, ysn+1, be the weighted average of all previously observed values yi, where the weight of yi is the similarity between the vector x1n+1,…,xmn+1, associated with yn+1, and the previously observed vector, x1i,…,xmi. 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.