###
UNBIASED BAYES ESTIMATES AND IMPROPER PRIORS

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

^{1} *Dipartimento di Economia, Politica e Metodi Quantitativi, Università di Pavia,*

Via S. Felice, 5, I-27100 Pavia, Italy

^{2} *Dipartimento 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.

**Source**
( TeX ,
DVI ,
PS )