  The problem of forecasting conditional probabilities of the next event given
the past is considered in a general probabilistic setting. Given an arbitrary
(large, uncountable) set C of predictors, we would like to construct a single
predictor that performs asymptotically as well as the best predictor in C, on
any data. Here we show that there are sets C for which such predictors exist,
but none of them is a Bayesian predictor with a prior concentrated on C. In
other words, there is a predictor with sublinear regret, but every Bayesian
predictor must have a linear regret. This negative finding is in sharp contrast
with previous results that establish the opposite for the case when one of the
predictors in $C$ achieves asymptotically vanishing error. In such a case, if
there is a predictor that achieves asymptotically vanishing error for any
measure in C, then there is a Bayesian predictor that also has this property,
and whose prior is concentrated on (a countable subset of) C.
