A Note on Unbiased Bayes Estimates

Suppose x and y are random variables that satisfy both E(x|y) = y and E(y|x) = x.If we think of x as an observation and y as a parameter, then the first relation says that the parameter y is the mean of x and the second says that the Bayes estimate of this parameter (i.e., the posterior mean of the parameter, given the observation) is unbiased.We show that provided either (a) E|x| is finite or (b) x is non-negative, all of the probability must be concentrated on the set {x = y}; but if both (a) and (b) fail then this conclusion does not follow.We give an explicit counterexample.If the joint distribution is allowed to be improper (but with proper conditionals), then (b) does not imply the conclusion; we give both continuous and discrete counterexamples.If the support of either x or y has no point of accumulation, however, the conclusion does follow.