Causal graphs and counterfactuals

[Stuart_Armstrong]

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 $A=1$).

Suppose 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 $P\left(B|A\right)$ 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:

• $f_0\left(x\right)=0$

• $f_1\left(x\right)=1$

• $f_2\left(x\right)=x$

• $f_3\left(x\right)=1-x$

Then one way of modelling the causal graph is as a mix $0.75f_2+0.25f_3$. In that case, knowing that $B=0$ when $A=0$ implies that $P\left(f_2\right)=1$, so if $A=1$, we know that $B=1$.

But we could instead model the causal graph as $0.5f_2+0.25f_1+0.25f_1$. In that case, knowing that $B=0$ when $A=0$ implies that $P\left(f_2\right)=2/3$ and $P\left(f_0\right)=1/3$. 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 $(0.5+x)f_2+x\left(f_3\right)+(0.25-x)f_1+(0.25-x)f_0$ for $0\le x\le 0.25$). 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?

[IAFF-User-52]

The problem is indeed that $P\left(B|A\right)$ is insufficient to compute a unique counterfactual—additional causal information is needed. Pearl’s approach is to specify each observable variable as a deterministic function of its parents in the causal graph. Any uncertainty must be represented by a set of “exogenous” variables $U$, which can feature in the functions for the observables. (See chapter 7 of Causality, or also An Axiomatic Characterization of Causal Counterfactuals.)

For example, your first process could be represented by the following causal model:

$A\left(X\right)=XB(A,Y)=\neg (A\oplus Y)P\left(X\right)=pP\left(Y\right)=0.75$

The other processes might have different structures, equations, and distributions—$P(X,Y)$it’s not possible in general to distinguish these purely from the distribution $P(A,B)$.

[Stuart_Armstrong]

Thank you! That sentence is what I was looking for “Any uncertainty must be represented by a set of “exogenous” variables U”.

I’d been doing that, but without any theoretical justification for it.