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I am trying to explain in simple words the Causal Markov condition to establish probabilistic causation. The original definition (Hausman and Woodward 1999) is the following:

“Let G be a causal graph with vertex set V and P be a probability distribution over the vertices in V generated by the causal structure represented by G. G and P satisfy the Causal Markov Condition if and only if for every X in V, Y is independent of V\(Descendants(X) ∪ Parents(X)) given Parents(X)”

My explanation is that a Causal Markov condition is satisfied if the set of variables in a causal relationship with given probability distributions are independent of all the other variables unless they are their parents or their descendants. This is slightly different than other definitions around so, is my explanation first, correct?, second clear? Any advice will be appreciated.

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  • $\begingroup$ You have omitted the last clause from Hausman and Woodward's definition: "given Parents(X)". Without that clause, both the definition and your restatement of it are incorrect. $\endgroup$ – Lizzie Silver Aug 20 '16 at 21:06
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    $\begingroup$ Note also that Hausman and Woodward (1999) did not originally define the Causal Markov Condition. They clearly cited Spirtes, Glymour and Scheines' book Causation, Prediction and Search, and note that the CMC was "apparently first described by Kiiveri and Speed (1982)". $\endgroup$ – Lizzie Silver Aug 20 '16 at 21:12
  • $\begingroup$ Y doesn't seem to be quantified anywhere in the definition. $\endgroup$ – Kodiologist Aug 20 '16 at 21:21
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    $\begingroup$ Just got both the Kiiveri paper and the Spirtes, Scheines and Glymour book. References corrected. Also added the last missing clause. Fortunately it was included in my original Ms. Thanks for noticing. $\endgroup$ – LDBerriz Aug 20 '16 at 21:59
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One way to think about the Causal Markov Condition (CMC) is giving a rule for "screening off": once you know the values of $X$'s parents, all other variables in $V$ become irrelevant for predicting $X$, except for $X$'s descendants.

I find examples make the CMC easiest to understand. I did a quick google image search for "mechanism of cardiovascular disease" so I can give you a medical example. Take this graph (let's call it $G$):

mechanism of cardiovascular disease. Source: Nat Clin Pract Cardiovasc Med (2008) Nature Publishing Group

Say you have a probability distribution $P$ over the variables in $G$. If the CMC holds in $P$ (relative to $G$), then you can infer that:

  • If I know the patient's amount of Oxidative stress and inflammation, then learning the patient's degree of Plaque progression won't give me any extra information about the patient's Platelets.
  • If I know the patient's amount of Atheroma, then learning the amount of Oxidative stress and inflammation won't help me predict Plaque progression.

However, the CMC allows the following possibilities:

  • If I know the patient's amount of Atheroma, then learning the patient's degree of Plaque rupture might still tell me something more about Plaque progression. (I'm learning from a descendant variable.)
  • If I don't know the patient's amount of Oxidative stress and inflammation, then learning the patient's degree of Plaque progression might well give me some extra information that helps me predict the patient's Platelets. (I'm not conditioning on the parents.)
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  • $\begingroup$ Thanks for the helpful example. The First three points are clear. However, if I don't know Oxidative stress and inflammation, but I know about Plaque progression but I am not conditioning on the parent Atheroma do I learn about Platelets form the common descendant Myocardial ischemia and infarction _ ? And if that is the case, would I also learn about _Endothjelium ? $\endgroup$ – LDBerriz Aug 20 '16 at 22:41
  • $\begingroup$ You're asking whether information can be transmitted between Atheroma and Platelets through the common child, Myocardial ischemia and infarction. But you're giving an example where information will be transmitted through the common parent, Oxidative stress and inflammation, because you aren't conditioning on it. Say we condition on Oxidative stress and inflammation so we block that "back-door" flow of information. Then by the CMC, Atheroma and Platelets will be independent. So you see how the CMC rules out transmitting information through a common child. $\endgroup$ – Lizzie Silver Aug 20 '16 at 22:56
  • $\begingroup$ Another even simpler example: say the graph is just $A \to C \leftarrow B$. Then the parent sets of $A$ and $B$ are the empty set, so you're always conditioning on their parents (trivially). Because neither $A$ nor $B$ is a descendant of the other, they are independent by the CMC, even though they have a common child $C$. $\endgroup$ – Lizzie Silver Aug 20 '16 at 22:58
  • $\begingroup$ Now clear. "Conditioning" blocks the flow of new information because you already know about it. Thanks for your help. $\endgroup$ – LDBerriz Aug 20 '16 at 23:33

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