# Understanding the conditional independence rule in chains

Pearl et al. "Causal Inference in Statistics: A Primer" (2016) p. 39 states the following:

Rule 1 (Conditional Independence in Chains) Two variables, $$X$$ and $$Y$$, are conditionally independent given $$Z$$, if there is only one unidirectional path between $$X$$ and $$Y$$ and $$Z$$ is any set of variables that intercepts that path.

(And then notes that the rule only holds if the error terms associated with the variables are independent of each other.)

As a non-native speaker, I am not entirely sure I understand the rule correctly. Does the if clause say that

1. there can only be one path AND
2. that path must be unidirectional?

Or does it say that

1. there can be many paths, but
2. among them only a single one is unidirectional?

Or ...? (My understanding of the English punctuation suggests the second alternative, but my understanding of the context points to the first one.)

Here's an example that casts doubt on your second interpretation, but is compatible with the first one. Consider the following R code.

a = runif()
b1 = a + runif()
b2 = a + runif()
c1 = b1 + runif()
c2 = b2 + runif()
d = c1 + c2 + runif()


This corresponds to the following DAG.

a  -> b1 -> c1
↓           ↓
b2 -> c2 -> d


Suppose we are assessing the independence of A and C1 conditional on {B1,D} as claimed by this lemma or definition. By the second understanding, the criterion is satisfied: there is only one unidirectional path, a->b1->c1, and it is interrupted. By the first understanding, the criterion is not satisfied, because there exists another path a-> b2 -> c2-> d<-c1 (though it is not unidirectional).

Suppose D is 0.1 for this whole example. If C1 is also 0.1, then A must be 0, because $$0=D-C_1=C_2\geq A \geq 0$$. Under smaller values of C1, A may be as large as 0.1. So in terms of probability theory, A and C1 are not independent conditional on D. Thus, either the lemma describes a concept of conditional independence that is distinct from that typically used in probability theory, or (more likely) the first understanding, not the second, is what Pearl meant -- though you're right that the wording matches the second better.

• I may need to modify the details of this little example, because my initial idea incorrectly asserted that 0 and 0 are not independent. Clearly they are. I think the same DAG will work somehow though. Apr 14, 2020 at 19:35
• Right, conditioning on $D$ opens up the collider. You could coalesce $B_2$ and $C_2$ to simplify a bit. Apr 14, 2020 at 20:04
• Another couple of related threads (in case you had time for them): "Adjustment formula for counterfactuals: can we get rid of $X=x$?" and "Law of total probability and conditioning on multiple events" (just to confirm/disconfirm the existing answer). Apr 15, 2020 at 17:30

I think Pearl is a bit ambiguous. (Thanks to eric_kernfeld for improving my understanding.) From the point-of-view of normal English usage, it is the second understanding. The adjective "unidirectional" modifies the first occurrence of the word "path", making up a single term: unidirectional path. The "if" part says there is only one of those. To say the first, you would have to word it like this:

... if there is only one path, unidirectional, such that...

or

... if there is only one path, that path is unidirectional, and ...

On the other hand, this DAG shows that $$X$$ and $$Y$$ can be dependent, even if $$Z$$ satisfies the second interpretation. Here, $$X$$ and $$Y$$ are not independent conditional on $$Z,$$ even though $$Z$$ satisfies the second interpretation.

In context, the first interpretation makes more sense.

• Interestingly, a previous answer (now deleted) said Your first understanding is correct. You can generalize Rule 1 to multiple unidirectional paths, as long as the variables in Z intercept all of those paths (and error terms are orthogonal). Now a quick follow-up question: why cannot there be many unidirectional paths where $Z$ would intercept each of them? This would be a less restrictive definition. Is the latter not restrictive enough so that only the definition as given is valid? (I can later post it separately if you have an answer so that you earn all the points you deserve.) Apr 14, 2020 at 18:59
• @eric_kernfeld, could you please elaborate? Apr 14, 2020 at 19:01
• Reposting my deleted comment: it said "The rule doesn't say if and only if". Apr 14, 2020 at 19:02
• @eric_kernfeld I'm not sure that matters in this context. Apr 14, 2020 at 19:03
• @RichardHardy I think in the spirit of the Rule, that would likely also be sufficient conditions for conditional independence. Apr 14, 2020 at 19:04