Is there a probabilistic graphical model for this situation?

Question

Suppose we have $3$ random variables, $X, Y, Z$, which verify: $$I(X,Y), I(X,Z), I(Y,Z),$$ where $I(·,·)$ means the variables are independent, and $$D(Y,X \ | \ Z), $$ meaning $Y$ and $X$ are dependent given $Z$.

It is clear not a Bayesian Network neither a Markov Network can model this situation, with the problem mainly coming from the fact that these variables are independent to each other, and thus we cannot place any (directed or undirected) arc between them.

Choosing the empty DAG or graph misses the dependency above. A reasonable Bayesian Network is $Z \to X \leftarrow Y$. This v-structure guarantees it is not losing the information $D(Y,X \ | \ Z)$, but it is assuming $D(Z,X)$ and $D(X,Y)$. At least this can indeed be corrected by using the adequate parameters in the Conditional Probability Tables, i.e. making $P(X \ | \ Z ) = P(X)$.

Now it makes no sense to use a Bayesian Network here, and also all Bayesian Network structure learning algorithms will simply fail (*) building this structure and construct an empty DAG instead.

So, What can be done? Is there a probabilistic graphical model for this situation?

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An example toy dataset descriptive of the situation above:

X Y Z
1 1 1
1 0 0
1 1 1
1 0 0
1 1 1
1 0 0
0 1 0
0 0 1
0 1 0
0 0 1
0 1 0
0 0 1

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(*) (edited) This is not necessarily true. Actually, it depends on many factors, mainly the depth of the search, meaning it can, indeed, capture the dependency $D(Y,X | Z)$ and thus place the needed arcs. They will still find trouble detecting these types of dependencies in general graphs (this is a toy example)

Answer 1

The statements that the three variables $ X $, $ Y $ and $ Z $ are pairwise independent is incompatible with the statement that $ X $ and $ Y $ are somehow dependent.

However, the toy dataset seems to indicate some kind of phenomenon where the presence of $ Z $ implies correlation between $ X $ and $ Y $, and the lack of $ Z $ makes them "less correlated" in some sense.

I think you might be slightly confused about the concept of independence, because in this case $ X $, $ Y $, and $ Z $ are certainly dependent. An example of a graphical model that could describe this is:

$$ A \to B $$

where $ A $ is a Bernoulli random variable (which corresponds with the $ Z $ in your model) and $ B $ is a bivariate Bernoulli random variable (i.e. it's a random vector of length 2). In this case, you could define the covariance of $ B_1 $ and $ B_2 $ in terms of $ A $.