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SilasBarta comments on Causality does not imply correlation - Less Wrong
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And in the current comment section, I'm going to give away the answer, since I've run through the PCT demos. (Sorry, I don't know how to format for spoilers, will edit once I figure out or someone tells me.)
You sure you didn't want to figure out on your own? Okay, here goes. Kennaway is describing a feedback control system: a system that observes a variable's current value and outputs a signal that attempts to bring it back towards a reference value. A is an external disturbance. B is the deviation of the system from the reference value (the error). C is the output of the controller.
The controller C will push in the opposite direction of the disturbance A, so A and C will be about anti-correlated. Their combined effect is to keep B very close to zero with random deviations, so B is uncorrelated with both.
The disturbance and the controller jointly cause the error. So, we have A->B and C->B. The error also causes the controller to output what it does, so B->C. (I assume directed cycles are allowed since there are four possible connections and you said there are 16 possible graphs.)
Together, that's A-->B<-->C
(In other news, Kennaway or pjeby will suggest I'm not giving due attenction to Perceptual Control Theory.)
(Edit: some goofs)
You have read my mind perfectly and understood the demos! But I'll go ahead and make the post anyway, when I have time, because there are some general implications to draw from the disconnect between causality and correlation. Such as, for example, the impossibility of arriving at A-->B<-->C for this example from any existing algorithms for deriving causal structure from statistical information.
Correct me if I'm wrong, but I think I already know the insight behind what you're going to say.
It's this: there is no fully general way to detect all mutual information between variables, because that would be equivalent to being able to compute Kolmogorov complexity (minimum length to output a string), which would in turn be equivalent to solving the Halting problem.
You're wrong. :-)
Kolmogorov complexity will play no part in the exposition.
Check my comment: I was only guessing the underlying insight behind your future post, not its content.
I obviously leave room for the possibility that you'll present a more limited or more poorly-defended version of what I just stated. ;-)