[Thanks to Matthias Georg Mayer for pointing me towards ambiguous counterfactuals]
Salary is a function of eXperience and Education
S=aE+bX
We have a candidate C with given salary, experience (X=5) and education (E=5).
Their current salary is given by
S=a⋅5+b⋅5
We ’d like to consider the counterfactual where they didn’t have the education (E=0). How do we evaluate their salary in this counterfactual?
This is slightly ambiguous—there are two counterfactuals:
E=0,X=5 or E=0,X=10
In the second counterfactual, we implicitly had an additional constraint X+E=10, representing the assumption that the candidate would have spent their time either in education or working. Of course, in the real world they could also have dizzled their time away playing video games.
One can imagine that there is an additional variable: do they live in a poor country or a rich country. In a poor country if you didn’t go to school you have to work. In a rich country you’d just waste it on playing video games or whatever. Informally, we feel in given situations one of the counterfactuals is more reasonable than the other.
Coarse-graining and Mixtures of Counterfactuals
We can also think of this from a renormalization / coarsegraining story. Suppose we have a (mix of) causal models coarsegraining a (mix of) causal models. At the bottom we have the (mix of? Ising models!) causal model of physics. i.e. in electromagnetics the Green functions give use the intervention responses to adding sources to the field.
A given counterfactual at the macrolevel can now have many different counterfactuals at the microlevels. This means we actually would get a probability dsitribution of likely counterfactuals at the top levels. i.e. in 1⁄3 of the cases the candidate actually worked the 5 years they didn’t go to school. In 2⁄3 of the cases the candidate just wasted it on playing video games.
The outcome of the counterfactual SE=0 is then not a single number but a distribution
SE=0=5⋅b+Y⋅b
where Y is random variable with distribution the Bernoulli distribution with bias 1/3.
Ambiguous Counterfactuals
[Thanks to Matthias Georg Mayer for pointing me towards ambiguous counterfactuals]
Salary is a function of eXperience and Education
S=aE+bX
We have a candidate C with given salary, experience (X=5) and education (E=5).
Their current salary is given by
S=a⋅5+b⋅5
We ’d like to consider the counterfactual where they didn’t have the education (E=0). How do we evaluate their salary in this counterfactual?
This is slightly ambiguous—there are two counterfactuals:
E=0,X=5 or E=0,X=10
In the second counterfactual, we implicitly had an additional constraint X+E=10, representing the assumption that the candidate would have spent their time either in education or working. Of course, in the real world they could also have dizzled their time away playing video games.
One can imagine that there is an additional variable: do they live in a poor country or a rich country. In a poor country if you didn’t go to school you have to work. In a rich country you’d just waste it on playing video games or whatever. Informally, we feel in given situations one of the counterfactuals is more reasonable than the other.
Coarse-graining and Mixtures of Counterfactuals
We can also think of this from a renormalization / coarsegraining story. Suppose we have a (mix of) causal models coarsegraining a (mix of) causal models. At the bottom we have the (mix of? Ising models!) causal model of physics. i.e. in electromagnetics the Green functions give use the intervention responses to adding sources to the field.
A given counterfactual at the macrolevel can now have many different counterfactuals at the microlevels. This means we actually would get a probability dsitribution of likely counterfactuals at the top levels. i.e. in 1⁄3 of the cases the candidate actually worked the 5 years they didn’t go to school. In 2⁄3 of the cases the candidate just wasted it on playing video games.
The outcome of the counterfactual SE=0 is then not a single number but a distribution
SE=0=5⋅b+Y⋅b
where Y is random variable with distribution the Bernoulli distribution with bias 1/3.