Gang Fang's Blog

Note 1

A Bayesian Network isn’t just a DAG, which I am often confused about, but a combination of two entities satisfying a relational condition. I can write it out as such: (G is a DAG and P is a probability density)

<G, P> where P satisfies the local directed markov condition for G

One important implication is you can’t say G is a Bayesian Network; you can’t say any pair of <G, P> is a Bayesian Network; but you can FIND Bayesian Networks given a G, if they exist.

Can you explain this hierarchy of equivalence relations between two different DAGs intuitively: markov equivalent, statistically equivalent and causally equivalent?

My answer: as we go from markov to statistical to eventually causal, the equivalence relation requires more rules and is thus harder to satisfy. Markov equivalence is concerned with G only, P is not considered. When statistical equivalence is discussed, we look at P(G), denoted as P represented by G, which consists of both P and G. Lastly, for two DAGs to be causally equivalent, they have to be statistically equivalent for every manipulation.

Can two DAGs with different structures be Markov equivalent?

My answer: imagine two almost DAGs, G and G’, which share the same base structure, A -> B -> C, but G’ has an additional node D whose parent is C (A -> B -> C -> D). It’s easy to see A and D is d-separated given C and this d-separated relation doesn’t exist in G. Therefore, two DAGs must share the same structure to be possibly Markov equivalent.

Critique: this answer is not correct because I didn’t regard different directions of edges as different structures. Two DAGs with the same nodes and connections (ignoring directions for a second) can still be different structurally when the directions are different, like (A -> B -> C) and (A <- B <- C).

In such cases, two DAGs with different structures can indeed have the same set of d-separated relations and hence be markov equivalent.