Title: Improper posteriors are not improper,
Abstract: In 1933 Kolmogorov constructed a general theory that defines the modern
concept of conditional expectation. In 1955 Renyi fomulated a new axiomatic
theory for probability motivated by the need to include unbounded measures. We
introduce a general concept of conditional expectation in Renyi spaces. In this
theory improper priors are allowed, and the resulting posterior can also be
improper.
In 1965 Lindley published his classic text on Bayesian statistics using the
theory of Renyi, but retracted this idea in 1973 due to the appearance of
marginalization paradoxes presented by Dawid, Stone, and Zidek. The paradoxes
are investigated, and the seemingly conflicting results are explained. The
theory of Renyi can hence be used as an axiomatic basis for statistics that
allows use of unbounded priors.
Keywords: Haldane's prior; Poisson intensity; Marginalization paradox;
Measure theory; conditional probability space; axioms for statistics;
conditioning on a sigma field; improper prior