Sufficiency and the Separation of Strongly Convex Sets of Probability Measures

We show that the equivalence of two notions of sufficiency is related to the separation of two orthogonal, strongly convex sets of probability measures by a universally measurable set. Consider a measurable space (äf, sé), and let M(%?) be the set of all probability measures on (%? , sé). Equip M(JT) with the canonical cr-algebra Jf generated by the functions m —► m(A), m in M(Sf), and A e se . A subset M of M(Sf) is said to be strongly convex if, for every probability measure p on M, the barycenter JM Pp(dP) is in M. Two strongly convex sets M and TV are orthogonal if for any P e M and Qe N there is a set Ap Q in A such that P(Ap q) = 1 and Q(Ap Q) 0. M and TV are said to be uniformly orthogonal if there is a set A in sé such that P(A) = 1 for all P in M and Q(A) = 0 for all ß in TV ; in such a case, A is said to separate M and TV. A natural question in this context is, "if two strongly convex sets M and TV are orthogonal, then are they uniformly orthogonal?" This question has received much attention in recent times, for instance see [8] and the references therein. We show in this article that the above question is equivalent to a problem arising in the study of "sufficiency" in mathematical statistics. Throughout this paper, we assume that (%?, sé) is a standard Borel space and that J'cj/ is a countably generated cr-algebra. If M is an analytic set of probability measures on (%?, sé), then (%?, sé , M) is called a standard Borel experiment. Let P be a probability measure on (3?, sé). For any bounded sé -measurable function /, the conditional expectation of / given 38 under P is a function g such that (i) g is ^-measurable and (ii) ¡B gdP = fB fdP for all B in S§. We shall denote by Ep(f\&) any version of the conditional expectation of / given 3§. Doob [4] defined a somewhat weaker notion of conditional expectation. According to Doob, ~g is a conditional expectation of / given ¿% if (i ' ) ~g is measurable with respect to the P-completion of ¿& and (ii ' ) ¡B gdP = JB fdP for all B in & . Received by the editors October 20, 1989 and, in revised form, December 18, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 62B20. ©1991 American Mathematical Society 0002-9939/91 $1.00+ $.25 per page