Finite exchangeability and linear regression

Abstract : We consider a finite sequence of exchangeable binary random variables and assume that the conditional expected value of each variable is an average of the observed frequency of success and a prior guess. We obtain a representation of the law of the finite sequence as a beta-binomial. In this way we prove known results for infinite sequences by a simple passage to the limit. Our approach does not have the generality of the usual representation theorems for (finite or infinite) exchangeable sequences, but is very natural and is completely elementary.