Let be the natural parameter space of the exponential family (2.35), and for any fixed tr+1,...,tk

Let Ω be the natural parameter space of the exponential family

(2.35), and for any fixed tr+1,...,tk (r

θ1...θr be the natural parameter space of the family of conditional distributions given Tr+1 = tr+1,...,Tk = tk.

(i) Then Ω

θ1,...,θr contains the projection Ωθ1,...,θr of Ω onto θ1,...,θr.

(ii) An example in which Ωθ1,...,θr is a proper subset of Ω

θ1,...,θr is the family of densities pθ1θ2 (x, y) = C(θ1, θ2) exp(θ1x + θ2y − xy), x, y > 0.

2.9 Notes The theory of measure and integration in abstract spaces and its application to probability theory, including in particular conditional probability and expectation, is treated in a number of books, among them Dudley (1989), Williams

(1991) and Billingsley (1995). The material on sufficient statistics and exponential families is complemented by the corresponding sections in TPE2. Much fuller treatments of exponential families (as well as sufficiency) are provided by Barndorff–Nielsen (1978) and Brown (1986)