# Can causal Bayes Nets compute counterfactuals? If so, are they “worse” than structural causal model counterfactuals?

I was under the impression that you couldn't perform counterfactual reasoning on causal Bayes nets using inference algorithms and do-calculus, but I realized recently that this might not be true. Does anyone know conclusively? If causal Bayesian networks can answer counterfactual queries, are they somehow inferior to those computed by structural causal models?

• This may be because of my lack of expertise in this field, but what exactly do you mean by 'counterfactual reasoning' in this context? Can you give an example in a concrete setting of the kind of question you'd like to have answered by Bayesian Networks? – Maurits M Nov 7 '18 at 9:42

## 1 Answer

By definition, Causal Bayesian Networks (CBN) encode only interventional distributions (you can check definition 1.3.1 of Causality), and involved counterfactuals such as $$P(y_{x} \mid x', y')$$, cannot be computed (nor even defined) in CBNs (in english, this is the probability that $$Y$$ would have been $$y$$ had $$X$$ been $$x$$, given that $$Y$$ was in fact $$y'$$ and $$X$$ was in fact $$x'$$).

To see this in practice, you can create counterexamples where we have two different structural causal models that induce the same observational and interventional probability distributions (the same DAG), yet induce different counterfactual distributions. In fact, we have such examples here on CV, you can check them here and here. And also a more extensive discussion between the difference of interventions and counterfactuals here.