  Laplace's "add-one" rule of succession modifies the observed frequencies in a
sequence of heads and tails by adding one to the observed counts. This improves
prediction by avoiding zero probabilities and corresponds to a uniform Bayesian
prior on the parameter. The canonical Jeffreys prior corresponds to the
"add-one-half" rule. We prove that, for exponential families of distributions,
such Bayesian predictors can be approximated by taking the average of the
maximum likelihood predictor and the \emph{sequential normalized maximum
likelihood} predictor from information theory. Thus in this case it is possible
to approximate Bayesian predictors without the cost of integrating or sampling
in parameter space.
