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A related question is here. As far as I can understand from scanning reviews on causal discovery, there are two critical conditions, (1) the causal markov condition and (2) causal faithfulness.

It is not clear to me that either of these conditions fails for just 'standard' Bayesian networks, and, how the PC/FCI algorithms or other algorithms out there, constrain the search space for Bayesian networks with a causal interpretation. Concretely then, my question would be: How does the PC algorithm find Bayesian networks with a causal interpretation, versus just any algorithm to find Bayesian networks in general? We know that we can find multiple Bayesian networks that satisfy conditional independence relations, but what allows us to interpret the output of PC algorithms to have that causal interpretation?

Additionally, this example on 3 nodes clarified for me that you can have several Bayesian networks, but only one which has certain values in the interventional probabilities that can be checked against data. Still, I am confused as to how causal discovery algorithms encode these interventional probabilities (or something equivalent to it) to check against data.

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  • $\begingroup$ I think I've figured out one case. Let's take the case of three variables A, B, C in the PC algorithm. Suppose we obtained the skeleton A -- C -- B, then the PC algorithm dictates that the shape should be A-> C <- B, because otherwise A would have been independent on B conditioned on C (assuming no unmeasured confounders or other variables). So the fundamental part is the choice of direction-setting rules that are compatible with a 'causal' interpretation. Same with the case if A -> B -- C, where it requires that A -> B -> C, which is the only directions compatible with a causal interpretation $\endgroup$
    – Pallen
    Commented Oct 22, 2020 at 18:00
  • $\begingroup$ Is it the case that non-causally based algorithms that generate Bayesian networks are not able to distinguish between 'causally meaningful' arrows and merely 'conditional independence-encoding' type arrows? It seems like one advantage of the PC and other algorithms is that it is able to label certain arrows as 'uncertain' in terms of causal direction, whereas a 'merely' Bayesian network algorithm would encode it as a definite arrow. $\endgroup$
    – Pallen
    Commented Oct 22, 2020 at 18:45
  • $\begingroup$ Are there Bayesian network algorithms that can generate Bayesian networks that do not have a causal interpretation? Or is it that non-causal Bayesian algorithms would not be able to distinguish which arrows are causal or merely conditional, and won't output that? Are there cases where you don't have to use a causal algorithm? It seems like causal algorithms are strictly better than a Bayesian algorithm. $\endgroup$
    – Pallen
    Commented Oct 22, 2020 at 18:53

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