New error bounds for Solomonoff prediction

Solomonoff sequence prediction is a scheme to predict digits of binary stri ngs without knowing the underlying probability distribution. We call a pred iction scheme informed when it knows the true probability distribution of t he sequence. Several neu relations between universal Solomonoff sequence pr ediction and informed prediction and general probabilistic prediction schem es will be proved. Among others. they show that the number of errors in Sol omonoff prediction is finite for computable distributions, if finite in the informed case. Deterministic variants will also be studied. The most inter esting result is that the deterministic variant of Solomonoff prediction is optimal compared to any other probabilistic or deterministic prediction sc heme apart from additive square root corrections only. This makes it well s uited even for difficult prediction problems, where it does not suffice whe n the number of errors is minimal to within some factor greater than one. S olomonoff's original bound and the ones presented here complement each othe r in a useful way. (C) 2001 Academic Press.