  We consider the Minimum Description Length principle for online sequence
prediction. If the underlying model class is discrete, then the total expected
square loss is a particularly interesting performance measure: (a) this
quantity is bounded, implying convergence with probability one, and (b) it
additionally specifies a `rate of convergence'. Generally, for MDL only
exponential loss bounds hold, as opposed to the linear bounds for a Bayes
mixture. We show that this is even the case if the model class contains only
Bernoulli distributions. We derive a new upper bound on the prediction error
for countable Bernoulli classes. This implies a small bound (comparable to the
one for Bayes mixtures) for certain important model classes. The results apply
to many Machine Learning tasks including classification and hypothesis testing.
We provide arguments that our theorems generalize to countable classes of
i.i.d. models.
