Implement Cyclic IDC

[cthoyt]

This PR proposes a cyclic variant of IDC building off the Cyclic ID algorithm implemented in #355

1. Generalize rule 2 of do-calculus to use sigma separation (works in cyclic graphs) instead of d-separation
2. Implement Cyclic IDC using the same application of rule 2 of do-calculus, but use the cyclic ID on the base case instead of regular ID

Question for @djinnome: is there already a paper proposing a cyclic IDC algorithm? If not, is this theoretically sound? Can we propose a toy example where this makes sense?

cc @emilydogbatsePNNL2025

[emilydogbatsePNNL2025]

this is an interesting thought. would love to hear what @djinnome thinks!

[djinnome]

The sigma calculus is a generalization of the do calculus for cyclic graphs, so it stands to reason that if we transform the consolidated districts of a cyclic graph to an equivalent ADMG, then in theory, IDC should work just fine. Appendix A of the cyclic ID paper strongly suggests that the transformation should also work for ID* and IDC*, too. Perhaps we should reach out to Forre and Mooij to see what they think?

[cthoyt]

hi @patrickforre @jorismooij, just reaching out because we've been working on implementing some of your ideas for identification on DMGs containing cycles in this repository ($Y_0$ Causal Inference Engine)

have you considered extension on the cyclic ID algorithm towards causal queries containing conditionals, similarly to how IDC extends ID?

[jorismooij]

Hi Charles, Great to learn that you're implementing some of our research! We have not yet thought about how to extend the IDC algorithm to deal with cycles. My guess is that the same ideas can be used, but working out the details is not something we have done yet. Best, Joris

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On Tue, Jan 06, 2026 at 12:43:15AM -0800, Charles Tapley Hoyt wrote: ●cthoyt left a comment (y0-causal-inference/y0#361) hi @patrickforre @jorismooij, just reaching out because we've been working on implementing some of your ideas for identification on DMGs containing cycles in this repository ($Y_0$ Causal Inference Engine) have you considered extension on the cyclic ID algorithm towards causal queries containing conditionals, similarly to how IDC extends ID? — Reply to this email directly, view it on GitHub, or unsubscribe. You are receiving this because you were mentioned.*Message ID: ***@***.***>