# Aren't $h$ and $t_1,t_2$ unconditionally independent?

I am currently reading about Independence Maps in Bayesian Reasoning and Machine Learning by David Barber and I'm trying to understand example 4.4, which is:

Example 4.4 (page 77) Consider the distribution (class) defined on variables $$t_1,t_2,y_1,y_2$$: $$p(t_1,t_2,y_1,y_2)=\\ p(t_1)p(t_2)\sum_h p(y_1|t_1,h)p(y_2|t_2,h)p(h)$$ In this case the list of all independence statements (for all distribution instances consistent with $$p$$) is $$\mathcal{L}_P=\{y_1 \perp\kern-5pt\perp t_2 | t_1, y_2 \perp\kern-5pt\perp t_1 | t_2, t_2 \perp\kern-5pt\perp t_1 \}$$

The belief network for the distribution $$(t_1,t_2,y_1y_2,h)$$ is (p 52):

Now, by what I understand, when we are talking about the distribution $$(t_1,t_2,y_1,y_2)$$, we marginalize $$h$$ and the belief network becomes:

Question 1: So the edge connecting $$y_1$$ and $$y_2$$ is bidirectional, (right?) And so this means that $$y_1 \perp\kern-5pt\perp t_2 | t_1$$ because $$t_1$$ is not a collider for $$y_1$$ and $$t_2$$ (similar reasoning for the other independence statements). But we have that $$y_1 \perp\kern-5pt\perp t_2 | y_2$$ is not true because $$y_2$$ is a collider, right?

Question 2: The example in the book says: Consider the graph of the BN $$p(y_2|y_1,t_1,t_2)p(y_1|t_1)p(t_1)p(t_2)$$ For this we have $$\mathcal{L}_G = \{y_1 \perp\kern-5pt\perp t_2 | t_1, t_2 \perp\kern-5pt\perp t_2\}$$. Now, I don't understand why the BN is this: This is a different graph from what I thought (the one above). How did they come up with this BN?

• I think your interpretation is reasonable, but technically I don't see how your first equation (and your belief network) follows from the given equation with the $\sum_h$. The author seems to be trying to treat $h$ in a different way from the other variables.
– Karl
Jul 6, 2022 at 16:41

I completely agree and the explanation is correct. One single point: I don't know about how Barber is doing it, but often the edge between $$y_1$$ and $$y_2$$, which you called "bidirectional", is depicted as $$\leftrightarrow$$. Thus it becomes "visible" that e.g. $$y_2$$ is a collider on the path from $$y_1$$ to $$t_2$$. (This can actually be taken further to arrive at a special kind of graph, called MAG (Maximal Ancestral Graph), which then has the nice property that marginalizations of perfect maps stay perfect.)
Why has he picked this BN, in particular, to check whether it could be a perfect map? He doesn't say. Maybe, his reasoning is somehow like this: If $$\;t_1 \perp\kern-5pt\perp\kern-8pt/\;\; y_2 \;|\; y_1$$ then there should be an edge between $$t_1$$ and $$y_2$$ to ensure this conditional dependency. And he then points out that this attempt to find a perfect graph just doesn't work.
For what it's worth, his set $$\mathcal L_P$$ in (4.5.8) is, contrary to his assertion, not the list of all independence statements, e.g. $$y_1 \perp\kern-5pt\perp t_2$$ (i.e. unconditionally) is missing (at least in my copy of the book).