Actually, this question is more or less a duplicate of the one which I have asked on math.stackexchange two days ago.
I did not get any answer there but I think now here is a better place to ask this question since it is more about "the philosophy", not the calculations involved in the concept, which is what the "math" board likes more.
I am trying to gain a good understanding of Bayesian Networks and the first thing I want to understand exactly is how they are built. I see that these networks are built on conditional independence assumptions in order to simplify joint distributions and they are built commonly by using causal relationships since they imply conditional independence given the direct causes. What I still don't understand is why these networks assume a Directed Acyclic (DAG) structure?
There can be systems which contain circular causality relationships. For example, let's think of a hypothetical machine consisting of two parallel plates which rub together. We think of three variables "Heat", "Plate Area Expansion" and "Friction". Plate Area Expansion is the effect of the cause "Heat" and "Friction" is just the effect of the plate area expansion since larger area means larger amount of friction, in turn. But if we think of it, "Friction" also causes an increase in the heat level, so "Friction" is a direct cause of "Heat" as well. This circular causality ends up with the following diagram:
This is a Directed Cyclic Graph and violates acyclicity (DAG) assumptions on which the whole Bayesian Network idea is founded. So, how can be such systems with circular causes, feedbacks and loops represented with Bayesian Networks? It is surely not possible that the DAG theory does not support such systems because this "feedback" mechanism is a quite common and basic thing. But I just cannot see how those kinds of systems are represented with acyclic graphs and I am sure that I am missing something here. I want to know what I am just missing.
Best Answer
Just to add a little more clarity. this approach is sometimes called Temporal Bayesian Models. I have seen it being used in atleast one other situation of marketing mix models where today's marketing spend influences today's brand & revenue. Today's brand also influences tomorrow's brand & revenue and so on.