Eliminating nuisance parameters: two characterizations

Given a statistical model P = {P-theta : theta is an element of Theta) and a surjective function g : Theta --> Lambda, the problem of transforming P into a new model Q = {Q(lambda) : lambda is an element of Lambda) indexed by Lambda is investigated. Two characterizations are given for those models Q of the form Q(lambda) = integral P-theta pi(lambda)(d theta), where pi(lambda) is some probability such that pi(lambda)(g = lambda) = 1. The first is related to a geometric property of Q, while the second rests on the inferential implications of adopting Q. Also, in the first pi(lambda) is allowed to be finitely additive, while in the second pi(lambda) is sigma-additive. Finally, integrated likelihoods are revisited in light of the second characterization.