Trying to understand equations in Karl Friston article
As mentioned in the definitions section on page 2 of the paper, $D(q||p) = &lt;ln(q/p)&gt;_q$ is the Kullback-Leibler divergence or cross-entropy between two densities.

The reason for the negative logs is that is the convention when discussing entropy in the context of information theory.  This allows information to be combined additively. (https://en.wikipedia.org/wiki/Entropy_(information_theory)#Rationale)