Fiducial Distributions and Bayes' Theorem

SUMMARY x is a one-dimensional random variable whose distribution depends on a single parameter θ. It is the purpose of this note to establish two results: (i)The necessary and sufficient condition for the fiducial distribution of θ, given x, to be a Bayes' distribution is that there exist transformations of x to u, and of θ to τ, such that τ is a location parameter for u. The condition will be referred to as (A). This extends some results of Grundy’s (1956).(ii)If, for a random sample of any size from the distribution for x, there exists a single sufficient statistic for θ then the fiducial argument is inconsistent unless condition (A) obtains: And when it does, the fiducial argument is equivalent to a Bayesian argument with uniform prior distribution for τ. The note concludes with an investigation of (A) in the case of the exponential family.