We bound the future loss when predicting any (computably) stochastic sequence
online. Solomonoff finitely bounded the total deviation of his universal
predictor M from the true distribution mu by the algorithmic complexity of
mu. Here we assume we are at a time t>1 and already observed x=x1...xt.
We bound the future prediction performance on xt+1xt+2... by a new
variant of algorithmic complexity of mu given x, plus the complexity of the
randomness deficiency of x. The new complexity is monotone in its condition
in the sense that this complexity can only decrease if the condition is
prolonged. We also briefly discuss potential generalizations to Bayesian model
classes and to classification problems.Comment: 21 page