Trying to understand equations in Karl Friston article

Question

I am trying to understand a neuroscience article by Karl Friston. In it he gives three equations that are, as I understand him, equivalent or inter-convertible and refer to both physical and Shannon entropy. They appear as equation (5) in this article (PDF; DOI 10.1007/s00422-010-0364-z). Here they are:

• Energy minus entropy: $F = −{\ln p(\tilde s,Ψ|m)}q + {\ln q(Ψ|μ)}q$
• Divergence plus surprise: = $D(q(Ψ|μ)||p(Ψ|\tilde s,m)) − \ln p (\tilde s|m)$
• Complexity minus accuracy: = $D(q(Ψ|μ)||p(Ψ|m)) − {\ln p(\tilde s|Ψ,m)}q$

The things I am struggling with at this point are:

1. the meaning of the || in the 2nd and 3rd versions of the equations
2. the negative logs

Any help in understanding how these equations are actually what Fristen claims them to be would be greatly appreciated. For example, in the 1st equation, in what sense is the first term energy, etc.?

Answer 1

As mentioned in the definitions section on page 2 of the paper, $D(q||p) = <ln(q/p)>_q$ 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)

Answer 2

1. In physics, exponential of negative energy is often probability (e.g. in maximum entropy models).
2. || is a notation used in divergences (or info theory) where the order matters. That is $D(A||B) \neq D(B||A)$ as in KL divergence