  We introduce new definitions of universal and superuniversal computable
codes, which are based on a code's ability to approximate Kolmogorov complexity
within the prescribed margin for all individual sequences from a given set.
Such sets of sequences may be singled out almost surely with respect to certain
probability measures. Consider a measure parameterized with a real parameter
and put an arbitrary prior on the parameter. The Bayesian measure is the
expectation of the parameterized measure with respect to the prior. It appears
that a modified Shannon-Fano code for any computable Bayesian measure, which we
call the Bayesian code, is superuniversal on a set of parameterized
measure-almost all sequences for prior-almost every parameter. According to
this result, in the typical setting of mathematical statistics no computable
code enjoys redundancy which is ultimately much less than that of the Bayesian
code. Thus we introduce another characteristic of computable codes: The
catch-up time is the length of data for which the code length drops below the
Kolmogorov complexity plus the prescribed margin. Some codes may have smaller
catch-up times than Bayesian codes.
