# Causality: Models, Reasoning, and Inference: Diagram Question

I am self-studying Causality: Models, Reasoning, and Inference, by Judea Pearl, and there is a question I am particularly stumped on. It reads like this:

Problem Statement: Given this fragment of a Bayesian Network, add two variables to the network, $$Z$$ and $$W,$$ such that the following three conditions hold simultaneously:

1. $$Z$$ and $$X$$ are dependent given $$W,$$ and
2. $$Z$$ and $$U$$ are independent given $$W.$$
3. $$Z$$ and $$W$$ are ancestors of $$Y$$ but not of $$X.$$

My Work So Far: Because $$Z$$ and $$W$$ must be ancestors of $$Y$$ but not $$X,$$ there can be no arrows going into $$X$$ (from $$Z$$ or $$W$$). Because $$U$$ is an ancestor of $$X,$$ there can be no arrows going into $$U,$$ either. Likely, though, we will need arrows going into $$Y.$$ As I see it, there are essentially two possibilities: arrows going out of $$U,$$ or arrows going out of $$X.$$ As we need dependence on $$X$$ and not $$U,$$ I'm going to guess that we need arrows going out of $$X.$$ That means there are essentially four possibilities:

a satisfies 1 and 3, but not 2. b satisfies 2 and 3, but not 1. c satisfies 1 and 3, but not 2. And d satisfies 1 and 3, but not 2.

Do I need to look at $$U?$$ Or do you have other ideas?

• In a, if you remove the arrow from Z to Y, it will still be an ancestor of it (but not a parent) and 2 holds – CloseToC Apr 27 at 23:50
• Are you positive this is correct? I can't see any way that an association between $X$ and $Z$ could exist without that same association existing for $U$ and $Z$. It's impossible to block the pathway between $U$ and $X$, so any association that $X$ has with its non-ancestors, $U$ will also have. – Noah Apr 28 at 1:33
• @CloseToC: Are you sure? Conditioning on $W$ opens up the collider there, allowing the path $Z\to W\leftarrow X\leftarrow U,$ so that $Z$ and $U$ would be dependent, right? – Adrian Keister Apr 28 at 2:04
• @Noah: Well, in my experience with Pearl's (co-authored) book Causal Inference in Statistics: A Primer, he certainly wasn't above giving trick questions for exercises. It's possible there's no answer. This problem goes with the text, but is actually a separate file. You can get it from Pearl's website bayes.cs.ucla.edu/jp_home.html and click on the Causality link at the top; you can eventually navigate to his Viewgraphs and Homeworks for Instructors link, which is where I found the homework files. – Adrian Keister Apr 28 at 2:07
• As you have solved it, please post that as an answer (that is, in the answer box) so the Q so not linger on as unsolved. – kjetil b halvorsen Apr 29 at 22:34