# Understanding math notation in infoGAN paper

I'm reading this paper about mutual information in infoGAN infoGAN_paper_link and already have the code to run it. I pretty much found code for it which is fine and dandy except for the fact that I kinda don't understand some of the code in the cost function. So, I looked at the paper to dissect it for better understanding and came across some math notation that I don't understand (pic below). The usage of the notations I'm trying to figure out are how the "~" in the Expectation subscripts, "'", and "||" symbols are used.

This is what I think the notation means.

• "~" in the expectation subscript: the variable on the left of "~" can be any continuous value coming from whatever is to the right of the "~"
• "'" next to the "c" in P(c'|x): I have no clue. I thought those were symbols for derivates but that makes no sense for this equation so it's def not that.
• "||": I'm not sure. I only know of these symbols being used in Norms but that's obviously not the case here.

Screenshot of a formula from the paper

The actual code I was trying to figure out was this in PyTorch. It's the variational lower bound (mutual information term) in the cost (It's not the formulas in the screenshot above). However, I ran into the formulas in the screenshot first before getting to the formula calculations for the variational lower bound formula.

mutual_information_lower_bound = lambda c_true, mean, logvar: Normal(mean,logvar.exp()).log_prob(c_true).mean()


Thanks for the help!

• The notation $$D_\mathrm{KL}(P \| Q)$$ is pretty much standard for the Kullback-Leibler divergence. There is an interesting discussion on Mathematics SE on the reasons for the fairly unusual notation used for divergences.

• In general, $$x \sim G(z, c)$$ means "$$x$$ is a random variable distributed according to $$G(z, c)$$". In the subscript the author means to take the expectation with respect to the random variable $$x$$, supposing that $$x$$ has that distribution.

• If you have a variable, such as $$a$$, it is not unusual to call a related variable $$a'$$ (usually pronounced "a prime"). In this case $$c'$$ is just a random variable distributed according to $$P(c\mid x)$$, with the name $$c'$$ probably meant to remind you that the variable is in some way connected to $$c$$. In cases where derivatives are obviously not the right thing, this is probably what a prime symbol means.