  Various optimality properties of universal sequence predictors based on
Bayes-mixtures in general, and Solomonoff's prediction scheme in particular,
will be studied. The probability of observing $x_t$ at time $t$, given past
observations $x_1...x_{t-1}$ can be computed with the chain rule if the true
generating distribution $\mu$ of the sequences $x_1x_2x_3...$ is known. If
$\mu$ is unknown, but known to belong to a countable or continuous class $\M$
one can base ones prediction on the Bayes-mixture $\xi$ defined as a
$w_\nu$-weighted sum or integral of distributions $\nu\in\M$. The cumulative
expected loss of the Bayes-optimal universal prediction scheme based on $\xi$
is shown to be close to the loss of the Bayes-optimal, but infeasible
prediction scheme based on $\mu$. We show that the bounds are tight and that no
other predictor can lead to significantly smaller bounds. Furthermore, for
various performance measures, we show Pareto-optimality of $\xi$ and give an
Occam's razor argument that the choice $w_\nu\sim 2^{-K(\nu)}$ for the weights
is optimal, where $K(\nu)$ is the length of the shortest program describing
$\nu$. The results are applied to games of chance, defined as a sequence of
bets, observations, and rewards. The prediction schemes (and bounds) are
compared to the popular predictors based on expert advice. Extensions to
infinite alphabets, partial, delayed and probabilistic prediction,
classification, and more active systems are briefly discussed.
