Bayesian statistics and biased procedures

A comparison of Neyman's theory of interval estimation with the corresponding subjective Bayesian theory of credible intervals shows that the Bayesian approach to the estimation of statistical parameters allows experimental procedures which, from the orthodox objective viewpoint, are clearly biased and clearly inadmissible. This demonstrated methodological difference focuses attention on the key difference in the two general theories, namely, that the orthodox theory is supposed to provide a known average frequency of successful estimates, whereas the Bayesian account provides only a coherent ordering of degrees of belief and a subsequent maximization of subjective expected utilities. To rebut the charge of allowing biased procedures, the Bayesian must attack the foundations of orthodox, objectivist methods. Two apparently popular avenues of attack are briefly considered and found wanting. The first is that orthodox methods fail to apply to the single case. The second is that orthodox methods are subject to a typical Humean regress. The conclusion is that orthodox objectivist methods remain viable in the face of the subjective Bayesian alternative — at least with respect to the problem of statistical estimation.