  The Bayesian framework is ideally suited for induction problems. The
probability of observing $x_t$ at time $t$, given past observations
$x_1...x_{t-1}$ can be computed with Bayes' rule if the true distribution $\mu$
of the sequences $x_1x_2x_3...$ is known. The problem, however, is that in many
cases one does not even have a reasonable estimate of the true distribution. In
order to overcome this problem a universal distribution $\xi$ is defined as a
weighted sum of distributions $\mu_i\inM$, where $M$ is any countable set of
distributions including $\mu$. This is a generalization of Solomonoff
induction, in which $M$ is the set of all enumerable semi-measures. Systems
which predict $y_t$, given $x_1...x_{t-1}$ and which receive loss $l_{x_t y_t}$
if $x_t$ is the true next symbol of the sequence are considered. It is proven
that using the universal $\xi$ as a prior is nearly as good as using the
unknown true distribution $\mu$. Furthermore, games of chance, defined as a
sequence of bets, observations, and rewards are studied. The time needed to
reach the winning zone is bounded in terms of the relative entropy of $\mu$ and
$\xi$. Extensions to arbitrary alphabets, partial and delayed prediction, and
more active systems are discussed.
