Do edges in directed acyclic graph represent causality? I am studying Probabilistic Graphical Models, a book for self-study. Do edges in a directed acyclic graph (DAG) represent causal relations? 
What if I want to construct a Bayesian network, but I am not sure about the direction of arrows in it? All the data will tell me is the correlations observed, not the inter-linking between them. I know I am asking too much, as I am sure following chapters will address these issues, but it's just that I can't stop thinking about it.
 A: Many structure learning algorithms can only score competing structures up to their Markov equivalences and as a result it is impossible to learn a unique DAG for a Bayesian Network (BN) based solely on data, which makes the causality hypothesis questionable. Spirtes et al. term this issue as “statistical indistinguishability”, discussing it at length in their book.
I take the view that the edges in a DAG should mainly be interpreted as probabilistic dependencies that also lend insight into causal relationships. This is in line with the viewpoint of the proponents of 'causal' Bayesian Networks (including Judea Pearl) who defend that the probability distribution represented by a BN has an underlying causal structure. 
The take-home message is, there does not exist an overarching agreement on this issue. But I guess the viewpoint I shared above is a safer one.
A: I only draw a directed edge if I am happy to assume the relationship is causal. This assumption of course cannot be verified by observational data, but by formalizing a set of hypothesized causal relationships as a DAG, I can identify which variables to adjust for to make the best possible causal inferences about a given relationship in the graph. From my perspective, if the DAG is true (big if, especially the acylic bit) then observed relationships amongst the variables should look a certain way; but it is still a complete absraction, and I don't see the value of that abstraction if you add arrows that don't reflect hypothesized causal relationships. 
A: Yes, the edges in DAG represent causal relationships. Consider an edge that goes from $A\rightarrow B$, this implies that $A$ 'causes' $B$.
Also it is impossible to construct a unique Bayes network, given just the data as different notions may lead to construction of different graphs.
A good resource for learning more about this can be found here.
A: As Zhubarb said, there is no overarching agreement on this issue. So, I'll throw in one more perspective that hasn't been covered yet. For causal DAGs, the causal structure is often considered to be encoded by the absence of arrows. Under this framework, the arrows may be causal or not, but missing arrows must be strongly believed or known to be not causal. This may not be widely applicable to Bayesian Networks, but since you started your question more generally, I think it's worth noting.
Also, if you want to learn a network, it won't be able to tell you the direction of arrows, because association flows both ways along arrows. You have to make some assumptions about directionality or impose some information about temporal ordering.
