Sufficient conditions for a family of probabilities to be exponential.

We make the following statement precise under fairly weak conditions: in an experiment, if we summarize n statistically independent observations (xi,. .. ,x.) in n m < n real numbers (yi,. . .,ym), where y = z Jfj(xi) and the fj are given funci=1 tions, and if we assume we have lost no information by the summary, then the family of probabilities associated with the experiment must be an exponential family. Let (X,21, IPt:t & T}) be fixed, where T is a set, 2 is a sigma-algebra of subsets of X, and {P1} is a family of probabilities, which satisfy P,(A) = 0 if and only if P,'(A) = 0 for (t,t',A) C T X T X 21. We say that {Pi} is an exponential family if for a fixed to & T there are p + 1 real-valued functions c; on T and p real-valued Borel functions pj on X, [spj-1(B) C 21 when B c R is a Borel set], so that