> There are many generalizarions of directed causal networks to include constraints (see e.g., the books and papers of Rina Dechter) and cycles (typically taken as the equilibrium distribution of a Markov chain; for example https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/1467-9868.00340, https://jmlr.csail.mit.edu/papers/volume1/heckerman00a/heckerman00a.pdf, https://www.ijcai.org/Proceedings/13/Papers/161.pdf).
It is true that many have considered generalizations of directed causal networks to include cycles and constraints, and we are aware of the references you point out. (In fact, we cut a discussion of Heckerman's dependency networks to streamline the story.) Yet none of these papers provide a satisfying answer to what these models mean at a qualitative level. (Heckerman shows that "consistent" DNs capture the same distributions as undirected graphical models, but this characterization only applies to special structures.) For acyclic models, there is an obvious answer: the structure implies certain independencies. But what is the analogue for a cyclic model? What can you say about the stationary distribution of a Markov chain? To answer this question you have to be more precise about what Markov chain you're talking about. In order to define an equilibrium semantics as you suggest (or indeed, to even formally define a Markov Chain for a cyclic network in which the state is a joint distribution), it is necessary to make a structural choice, that in a sense, breaks the symmetry promised by a cyclic representation. This choice can be made in the form of a sampling order (as is an important point in the Heckerman (2000) and Poole&Crowley (2013) papers you reference), or a cut set (as in the Baier et. al. (2022) paper that we reference). Either choice amounts to a selection of qualitative information that is not present in the underlying graph, and often swept under the rug. It is not hard to show that a choice of sampling order actually induces a *BN's* independencies, and therefore this approach does not say anything interesting about cyclic models at a qualitative level.
We also point out that Poole and Crowley paper you reference states that there "seem to be three solutions to causal modeling with cycles: (1) do not allow cycles, (2) make noise dependent, or (3) use a different (non-causal) semantics". Yet our approach uses causal semantics with independent noise, and allows for cycles!
> I thought the parity example was an obvious counter-example; because the variable was independent of each of the other variables, that it was clear. One theoretical justification is in terms of the Hadamard transform (or the discrete Fourier); one reference where this is applied to graphical models is https://proceedings.mlr.press/v22/buchman12.html The cycle loses the high frequency terms (the one needed for the parity term).
We too found the parity distribution to be an obvious candidate for a counter-example. Yet, as mentioned in our response, we found it (surprisingly) difficult to establish that there could be no witness satisfying the properties of our definition. While we understand that the Hadamard and discrete Fourier transforms are intimately related to parity systems, we do not see any way to apply them to demonstrate a lack of QIM-compatibility. Your intuition that a cycle should "lose high-frequency terms" concords with ours; indeed, such an argument can be used to show that the parity distribution $\mu_{\mathrm{xor}}$ cannot be written as $\mu_{\mathrm{xor}}(X,Y,Z) = f_1(X,Y) f_2(Y,Z) f_3(Z,X)$, for any choice of $f_1,f_2,f_3$. Yet despite having these intuitions, we still were very surprised how difficult it was to provide a formal proof that the parity distribution is not QIM-compatible (in the sense of Definition 2) with the 3-cycle.
Fortunately (in our opinion), the effort paid off in the general case: the information-theoretic test for QIM-compatibility with the 3-cycle is entirely novel, quite different from more standard spectral arguments (or those that rest on polynomial degrees), and has also helped to clarify the meaning of interaction information.