Given random variables $x, y, z$, is there any useful interpretation of the quantity $p(x|y)p(y|z)p(z|x)$?
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It doesn't have a useful interpretation, though what you're doing has a long history as an approximation.
It was introduced as a way to approximate the joint distribution $p(x, y, z)$ by Besag (1974). It's sometimes called the pseudolikelihood. Your particular expression also requires some conditional independence assumptions, but it works the same way; the pseudolikelihood is normally $p(x \mid y, z)p(y \mid z, x) p(z \mid x, y)$.
I first learned about this in an extra credit problem in Johns Hopkins University's natural language processing class. For the sake of students who take the class in the future, I won't go into details that might help them to answer that question.
answered May 16, 2021 at 0:29
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$\begingroup$ Interesting! I wonder what practical difference the additional conditions make. If you look at entropies instead of at probabilities, things start looking an awful lot like (parts of) the trivariate mutual information... $\endgroup$ Commented May 16, 2021 at 16:15