Unbiased Bayes estimates and improper prior

Given two random variables X and Y , the condition of unbiasedness states that E(X|Y = y) = y and E(Y |X = x) = x both almost surely (a.s.). If the prior on Y is proper and has finite expectation or nonnegative support, unbiasedness implies X = Y a.s. This paper examines the implications of unbiasedness when the prior on Y is improper. Since the improper case can be meaningfully analysed in a finitely additive framework, we revisit the whole issue of unbiasedness from this perspective. First we argue that a notion weaker than equality a.s., named coincidence, is more appropriate in a finitely additive setting. Next we discuss the meaning of unbiasedness from a Bayesian and fiducial perspective.We then show that unbiasedness and finite expectation of Y imply coincidence between X and Y , while a weaker conclusion follows if the improper prior on Y is only assumed to have positive support. We illustrate our approach throughout the paper by revisiting some examples discussed in the recent literature.