  Given two events $A$ and $B$, Bayes' law is based on the argument that the
probability of $A$ given $B$ is proportional to the probability of $B$ given
$A$. When probabilities are interpreted in the Bayesian sense, Bayes' law
constitutes a learning algorithm which shows how one can learn from a new
observation to improve their belief in a theory that is consistent with that
observation. Kolmogorov's notion of algorithmic information, which is based on
the theory of algorithms, proposes an objective measure of the amount of
information in a finite string about itself and concludes that for any two
finite strings $x$ and $y$, the amount of information in $x$ about $y$ is
almost equal to the amount of information in $y$ about $x$. We view this
conclusion of Kolmogorov as the algorithmic information version of Bayes' law.
This can be easily demonstrated if one considers the work of Levin on prefix
Kolmogorov complexity and then expresses the amount of Kolmogorov mutual
information between two finite strings using Solomonoff's a priori probability.
