Trying to understand the math in a neuroscience article by Karl Friston

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-convertertable and refer to both physical and Shannon entropy. They appear as equation (5) in the article at http://www.fil.ion.ucl.ac.uk/spm/doc/papers/Action_and_behavior_A_free-energy_formulation.pdf (DOI 10.1007/s00422-010-0364-z). Here they are:

• Energy minus entropy
• Divergence plus surprise
• Complexity minus accuracy

$$\begin{align*}F &=−\left\langle \ln p (\tilde{s},\Psi |m)\right\rangle_q+\left\langle \ln q(Ψ|\mu)\right\rangle_q \\ &= D\left(q(\Psi|\mu)\ ||\ p(\Psi|\tilde{s},m)\right)−\ln p(\tilde{s}|m) \\ &= D\left(q(\Psi|\mu)\ ||\ p(\Psi|m)\right)−\left\langle \ln p(\tilde{s}|\Psi,m)\right\rangle_q \end{align*}$$

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 each of these equations amounts to what Fristen describes it to be (at the left of the equation) would be greatly appreciated. For example, in the 1st equation, in what sense are the terms energy and entropy, etc? Is the entropy, Shannon or thermodynamic or both?

Answer 1

Not an answer, but it must be formatted in a way that a comment cannot hold.

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If you continue reading, the author defines this notation:

Here, $\langle \cdot \rangle_q$ means the expectation or mean under the density $q$ and $D(\cdot\ ||\ \cdot)$ is the cross-entropy or Kullback-Leibler divergence between the two densities.

I have taken liberties to embed links to the relevant Wikipedia articles in the quote