  Although Bayes's theorem demands a prior that is a probability distribution
on the parameter space, the calculus associated with Bayes's theorem sometimes
generates sensible procedures from improper priors, Pitman's estimator being a
good example. However, improper priors may also lead to Bayes procedures that
are paradoxical or otherwise unsatisfactory, prompting some authors to insist
that all priors be proper. This paper begins with the observation that an
improper measure on Theta satisfying Kingman's countability condition is in
fact a probability distribution on the power set. We show how to extend a model
in such a way that the extended parameter space is the power set. Under an
additional finiteness condition, which is needed for the existence of a
sampling region, the conditions for Bayes's theorem are satisfied by the
extension. Lack of interference ensures that the posterior distribution in the
extended space is compatible with the original parameter space. Provided that
the key finiteness condition is satisfied, this probabilistic analysis of the
extended model may be interpreted as a vindication of improper Bayes procedures
derived from the original model.
