UNBIASED BAYES ESTIMATES AND IMPROPER PRIORS

GUIDO CONSONNI^{1} AND PIERO VERONESE^{2}

1Dipartimento di Economia, Politica e Metodi Quantitativi, Università di Pavia,

Via S. Felice, 5, I-27100 Pavia, Italy 2Dipartimento di Economia Politica, Istituto di Metodi Quantitativi,

Università Commerciale Luigi Bocconi, Via U. Gobbi, 5, I-20136 Milano, Italy

(Received March 5, 1991; revised June 30, 1992)

Abstract. Given two random variables (X, 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 non-negative 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.

Key words and phrases: Coincidence, dF-coherence, equality almost surely, finite additivity, improper prior, unbiasedness.