counterfactual

• Level 3 of Pearl's causal inference hierarchy: questions of the form 'given that (X, Y) happened, what would have happened if (X', Y) had happened instead?'
• Source: Causal Inference 3: Counterfactuals
• The formal setup is that we
  1. copy the structural equation model to create a set of 'mirror world' variables,
  2. perform the desired intervention in the mirror world (mutilating the model to represent the intervention, as in do-calculus), and then
  3. share randomness $\varepsilon$ between the original and mirror-world models to define a joint distribution on both worlds. Since the worlds are coupled, we can now talk about conditional probabilities of mirror-world quantities given observed quantities.
• Note that although the original world and the mirror world are coupled, they are marginally just following the original and mutilated joint distributions, respectively.
  • If we want to ask, 'would Ferenc have gotten a PhD if he didn't have a beard, given that he did have a beard, is married, is in good shape, and did get a PhD'?, this is a question about Ferenc as an individual. If we average over all possible individuals (ie, all possible conditioning information, ie, the values in the original graph), we get back the population-level causal query p(PhD | do(beard=true)).

Another perspective on counterfactuals is that a