Revisiting Jeffreys’ Example: Bayes Test of the Normal Mean

We revisit the classical problem of testing whether a normal mean is zero against all possible alternatives within a Bayesian framework. Jeffreys showed that the Bayes factor for this problem has a drawback with normal priors for the alternatives. He showed also that this deficiency is rectified when one uses a Cauchy prior instead. Noting that a Cauchy prior is an example of a scale-mixed normal prior, we want to examine whether or not scale-mixed normal priors can always overcome the deficiency of the Bayes factor. It turns out though that while mixing priors with polynomial tails can overcome this deficiency, those with exponential tails fail to do so. Examples are provided to illustrate this point.