Lingzhou Michael Xue writes in with two questions:

1) Possible to Generalize the Rubin Causal Model?
In my undergraduate research project, I have discovered almost every subjects focused
on decoding the network-level causality in almost every field, ranging from Biology,
Medicine Design to even Social Science. However, these publications obviously lack solid
statistical foundations on the definition of causality and how to do causal analysis.

On the other hand, I have been enlightened a lot from Rubin Causal Model, and also
powerful tools such as instrumental variables and propensity scores. Yet, these causal
inference are limited in the one-variable causality. Is possible to generalize it to
deal with interaction causality? From my intuition, it seems pretty difficult to do this.
I am still curious about the possibility to generalize Rubin’s Model?

2) Some works on Bayesian Variable Selection?
Recently, we have witnessed fruitful and interesting reseraches on variable selection,
which even draw Terence Tao’s attention. What is more interesting, most works of this
area rely on penalized learning, i.e. from the frequentist perspective. While I believe
that Bayesian approach might bring us a more reasonable framework just as it always did.
could you kindly show me some works on bayesian variable selection?

My reply:

1. Rubin’s causal model allows for interactions. Interactions between treatment and pre-treatment predictors fit in automatically with no complications at all, except that the goal is no longer to estimate an average treatment effect, you now want to estimate the effect conditional on predictors. If you have interactions between different treatment factors, it just complexifies the potential outcomes. I agree, though, that when the treatment is continuous, the potential outcomes need to be modeled, which brings Rubin’s framework closer to classical regression and instrumental variables.

2. I’m not a big fan of variable selection. I prefer continuous model expansion: keeping all the variables in the model and controlling them with an informative prior distribution or hierarchical model.