Basically, making every endogenous variable a deterministic function of the exogenous variables and of the other endogenous variables, and pushing all the stochasticity into the exogenous variables.
Old post:
A problem that’s come up with my definitions of stratification.
Consider a very simple causal graph:
In this setting, A and B are both booleans, and A=B with 75% probability (independently about whether A=0 or A=1).
I now want to compute the counterfactual: suppose I assume that B=0 when A=0. What would happen if A=1 instead?
The problem is that P(B|A) seems insufficient to solve this. Let’s imagine the process that outputs B as a probabilistic mix of functions, that takes the value of A and outputs that of B. There are four natural functions here:
f0(x) = 0
f1(x) = 1
f2(x) = x
f3(x) = 1-x
Then one way of modelling the causal graph is as a mix 0.75f2 + 0.25f3. In that case, knowing that B=0 when A=0 implies that P(f2)=1, so if A=1, we know that B=1.
But we could instead model the causal graph as 0.5f2+0.25f1+0.25f0. In that case, knowing that B=0 when A=0 implies that P(f2)=2/3 and P(f0)=1/3. So if A=1, B=1 with probability 2⁄3 and B=1 with probability 1⁄3.
And we can design the node B, physically, to be one or another of the two distributions over functions or anything in between (the general formula is (0.5+x)f2 + x(f3)+(0.25-x)f1+(0.25-x)f0 for 0 ≤ x ≤ 0.25). But it seems that the causal graph does not capture that.
Owain Evans has said that Pearl has papers covering these kinds of situations, but I haven’t been able to find them. Does anyone know any publications on the subject?
Causal graphs and counterfactuals
Problem solved: Found what I was looking for in: An Axiomatic Characterization Causal Counterfactuals, thanks to Evan Lloyd.
Basically, making every endogenous variable a deterministic function of the exogenous variables and of the other endogenous variables, and pushing all the stochasticity into the exogenous variables.
Old post:
A problem that’s come up with my definitions of stratification.
Consider a very simple causal graph:
In this setting, A and B are both booleans, and A=B with 75% probability (independently about whether A=0 or A=1).
I now want to compute the counterfactual: suppose I assume that B=0 when A=0. What would happen if A=1 instead?
The problem is that P(B|A) seems insufficient to solve this. Let’s imagine the process that outputs B as a probabilistic mix of functions, that takes the value of A and outputs that of B. There are four natural functions here:
f0(x) = 0
f1(x) = 1
f2(x) = x
f3(x) = 1-x
Then one way of modelling the causal graph is as a mix 0.75f2 + 0.25f3. In that case, knowing that B=0 when A=0 implies that P(f2)=1, so if A=1, we know that B=1.
But we could instead model the causal graph as 0.5f2+0.25f1+0.25f0. In that case, knowing that B=0 when A=0 implies that P(f2)=2/3 and P(f0)=1/3. So if A=1, B=1 with probability 2⁄3 and B=1 with probability 1⁄3.
And we can design the node B, physically, to be one or another of the two distributions over functions or anything in between (the general formula is (0.5+x)f2 + x(f3)+(0.25-x)f1+(0.25-x)f0 for 0 ≤ x ≤ 0.25). But it seems that the causal graph does not capture that.
Owain Evans has said that Pearl has papers covering these kinds of situations, but I haven’t been able to find them. Does anyone know any publications on the subject?