From ordinal ordering to a prior probability

Billot et al. (Econometrica 73:1125–1136, 2005) (BGSS) propose a model for constructing a prior probability over states of nature based on past data. According to their model, the evaluator possesses a similarity function over observations and, given a database, adopts the prior that is the similarity-weighted frequency of outcomes within that database. BGSS thus simplifies the task of forming a prior probability over states, reducing it to the question of forming a similarity function over observations. Still, the task of creating a similarity function remains and may not always be straightforward. We characterize two relatively simple procedures for forming a similarity function, one which requires placing observations on an integer scale and the other which is essentially ordinal. These procedures could assist in the implementation of the BGSS model in certain real-life situations.