  Suppose we have $n$ different types of self-replicating entity, with the
population $P_i$ of the $i$th type changing at a rate equal to $P_i$ times the
fitness $f_i$ of that type. Suppose the fitness $f_i$ is any continuous
function of all the populations $P_1, \dots, P_n$. Let $p_i$ be the fraction of
replicators that are of the $i$th type. Then $p = (p_1, \dots, p_n)$ is a
time-dependent probability distribution, and we prove that its speed as
measured by the Fisher information metric equals the variance in fitness. In
rough terms, this says that the speed at which information is updated through
natural selection equals the variance in fitness. This result can be seen as a
modified version of Fisher's fundamental theorem of natural selection. We
compare it to Fisher's original result as interpreted by Price, Ewens and
Edwards.
