From: Arvid Sjolander, Dept. of Medical Epidemiology and Biostatistics, Karolinska Institutet

Subject:

**Question to author:**

At the bottom of page 214 you mention that "any variable
that is *d*-separated from *Z** would also be
*d*-separated from *U _{Z}*"
(and vice versa I guess?). You conclude that
"if

where

On the top of page 215 you use the twin network to show that

and then refer to the previous argument to state that

but how can you remove

If I would examine if (3) is true I would naively construct the following triple network:

The left part corresponds to the world in which no
intervention is imposed, the middle part in which *do*(*X=x*)
is imposed, and in the right part *do*(*Z=z*) is imposed. In
this network (3) does not hold since the path
is open by conditioning on *Y*.
**Is there anything wrong with this use of
the (generalized) twin network method?**

**Author's first reply (with Ilya Shpitser)**

You are, indeed, correct.

*Y _{x}* is not independent of

However, for example, *Y _{x}* is independent of

Your use of *d*-separation in the twin network to understand
independencies between underlying counterfactual quantities is correct,
and so is your generalization of the 'twin' network to more than
two possible worlds. We called this the 'parallel worlds model' in the
path-specific effects paper.

I have spoken to Judea about (3) last spring. Our conclusion was that
(3) is not true in general, but is true if the functions from *U*
to their children are one to one. I believe this is the case Judea
had in mind when he wrote that section of the book.

**Arvid's followup:**
1) I see that a one-to-one correspondence between *U _{v}*
and

Is there ever a reason for assuming the criteria would hold true?

2) As a bonus your answer helped me with a similar question on the
paper "Direct and Indirect Effects" (Pearl 2001), namely "why does (7)
hold in the graph in Figure 1(a)?". If one draws a triple graph it is
obvious that the path
is open. If however *U _{2}* is
completely determined by

**Reply to 1:**

I think many important classes of causal models exhibit one-to-one
relationships, for example structural equation models assume all functions
are linear, and so one-to-one. Frequently we also have causal models
that contain deterministic relationships between nodes which will result
in similar changes to the way *d*-separation works. I agree,
though, that the original discussion in Judea's book should make clear
it's talking about the one-to-one case.

**Reply to 2:**

(7) actually holds in the general case in Fig 1 (a).

This is because *Y _{xz}* is independent of

But we know by rule 3 of *do*-calculus that *W _{x*} = W*.

Next discussion (Yudkowsky: *The validity
of G-estimation*)

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