What Bayes Has to Say About the Evidence Procedure

The ``evidence'' procedure for setting hyperparameters is essentially the same as the techniques of ML-II and generalized maximum likelihood. Unlike those older techniques however, the evidence procedure has been justified (and used) as an approximation to the hierarchical Bayesian calculation. We use several examples to explore the validity of this justification. Then we derive upper and (often large) lower bounds on the difference between the evidence procedure's answer and the hierarchical Bayesian answer, for many different quantities. We also touch on subjects like the close relationship between the evidence procedure and maximum likelihood, and the self-consistency of deriving priors by ``first-principles'' arguments that don't set the values of hyperparameters.