Number of Causal Assumptions in an Overview by Pearl

Question

In the paper Causal Inference in Statistics: an Overview by Pearl, in page 11 (106 if you go by the Journal's indexing), a graphical model is presented in figure 2(a). The text reads (picture below):

The chain model of Fig. 2(a), for example, encodes seven causal assumptions, each corresponding to a missing arrow or a missing double-arrow between a pair of variables.

How did the author conclude there are seven missing arrows?

Answer 1

None of the below causal arrows appear in Fig. 2(a). I am assuming time flows from top left to bottom right (i.e. so that $Y \to X$ cannot be a causal assumption because causes must precede effects.).

1. $U_{Z} \to U_{X}$
2. $U_{Z} \to U_{Y}$
3. $U_{Z} \to X$
4. $U_{Z} \to Y$
5. $U_{X} \to U_{Y}$
6. $U_{X} \to Y$
7. $Z \to Y$

This means that the causal world in Fig. 2(a) assumes there are none of the above seven direct causal effects. By contrast, each of the arrows actually appearing in the graph (e.g., $U_{Z} \to Z$, etc.) are assumptions of direct causal effects.

EDIT: Based on correspondence with Judea Pearl. [Judea's quote is edited for the grammar/typos common in a brief email exchange.]

I had in mind the following

$U_{Z} \longleftrightarrow U_{X}$

$U_{Z} \longleftrightarrow U_{Y}$

$U_{X} \longleftrightarrow U_{Y}$

$Z \to Y$

$X \to Z$

$Y \to Z$

$Y \to X$

The missing arrows you listed e.g., $U_{X} \to Y$ are implied by the above, because $U_{Y}$ is defined as everything that affects $Y$ when $X$ is held constant.

Answer 2

An exchange of comments with @Alexis (and their correspondence with Pearl himself) cleared things up for me. I can summarize as follows:

1. For the exogenous variables $U_X, U_Y, U_Z$ we only allow/count double arrows (just... because?). For these variables we have three missing (double) arrows, which are $U_X \leftrightarrow U_Y, U_Z \leftrightarrow U_Y$ and $U_X \leftrightarrow U_Z$.

2. For the endogenous variables $X,Y,Z$, we count only directed arrows (again, just because) and we have four missing such arrows, which are $X\to Z, Y\to Z, Y \to X$ and $Z\to Y$.

3. We do not count arrows such as $U_X \to Z$ since $U_Z$ is defined as everything that affects $Z$ outside of the other endogenous variables ($X,Y$, in this case), so no other influence is allowed, specifically not $U_X$.

This count gives us seven missing arrows total, as the text suggests.