# Modeling non-acyclic reality with acyclic graphs

In my understanding, we use directed acyclic graphs to model causality in Bayesian networks. But it is common sense that there are feedback loops in life. How do we deal with this?

1. Reduce to strongly connected components ("feedback clusters"), abstract out the content interior to the clusters, and model the interaction between the clusters, using Bayes net techniques, do-calculus, path coefficients, etc.?
2. Use techniques adapted to non-acyclic graphs?

The correct answer to my question might be something like

There is no "how 'we' deal with this", at least not yet. Causal modeling is still developing, and there is no one good answer to this question. People have tried different modifications for different objectives, such as mixing undirected and directed graphs, or converting to an undirected moral graph.

Still, I'm surprised I haven't encountered any discussion of the matter as I'm reading the expository articles on the subject.

I'm interested in discussion of the principle (philosophy-adjacent), or discussion of the practice (math/stats).

• My understanding is that there are some results on this, but they are considerably more high-level, both to understand and to implement. Judea Pearl's Causality has some results along these lines, in Chapter 7.2.1, for example. He only handles a non-recursive linear system. He has a few valuable comments in section 3.6.1 you should check out, as well. Mar 20 '20 at 2:44

A simple solution is commonly encountered in epidemiology. It's based on the fact that causation and time go hand in hand. For a variable $$A$$ to "feedback" to itself, time must be involved. In this way feedback can be accounted for if we incorporate time into an acyclic graph.

For example, the following simple diagram demonstrates how a variable $$A$$ at time $$0$$ can can affect itself at time $$1$$, possibly through some mediating variable $$Z$$ measured at time $$1$$: The above figure is a still a directed acyclic graph, as it meets all the required criteria. It is a standard example of time varying confounding affected by prior exposure, of which a defining characteristic is feedback loops between nodes.

There is a fairly substantial body of literature in epidemiology and biostatistics on strategies for analyzing data generated from such structures. These include many DAGs that incorporate feedback loops. Here are three:

Naimi et al 2017

Robins and Hernan 2009

Daniel et al 2013

• Agreed. Common practice in epidemiology is indexing variables in a causal diagram by time. This has been done for a variety of time-scales (from days to months) Apr 7 '20 at 19:05