Pearl likes graphs, but graphs are just a mathematical aid. What he and Rubin are talking about is not “about” graphs. You can prove all the theorems without graphs. Both Pearl and Rubin are talking about potential outcomes (interventionist view of causality). Pearl uses a model which makes cross-world independence assumptions (Rubin probably does not, although I have not asked him. Of course Rubin loves “principal stratification” which as far as I understand is wildly untestable, so who really knows what he thinks. A lot of workers in the field do not like cross-world independences because they are not testable).
To the extent that Rubin wants to estimate potential outcome random variables from observational data, he HAS to agree with Pearl on pain of bias (e.g. garbage). In the example I gave, if Rubin insists on conditioning on W, he will get a garbage answer for the potential outcome Y(x). Identification of potential outcomes isn’t the kind of thing where you can have a difference of opinion. It’s like having on opinion on what 2 + 2 is.
In the example I gave, if Rubin insists on conditioning on W, he will get a garbage answer for the potential outcome Y(x).
From your description, you say that Rubin insists on conditioning on all available data, so that includes W. But that doesn’t mean he has to get garbage, that just means he needs the right conditional.
Let Jaynes notation do the work. The base problem seems to be:
You can assign probabilities using observational data to create P(X1...XN | Intervention=No). How do I use that model to assign P(X1...XN | Intervention=Yes)?
Do these guys have any case where they make different predictions of what will happen in an intervention? Or do they just dance around in their own languages and come up with the same predictions?
“From your description, you say that Rubin insists on conditioning on all available data, so that includes W. But that doesn’t mean he has to get garbage, that just means he needs the right conditional.”
The right expression for p(y | do(x)) in this example should ignore W, that’s all there is to it. It’s not a notational issue.
“You can assign probabilities using observational data to create P(X1...XN | Intervention=No). How do I use that model to assign P(X1...XN | Intervention=Yes)?”
Good question! The answer is to use something called the consistency assumption (I think Pearl might call it “composition” in his book). This states, roughly that Y(X) = Y. (That is, observing Y when there is no intervention is the same as observing Y when X is intervened to attain whatever value it would naturally attain). This assumption is untestable, but to my knowledge every single paper in causal inference makes this assumption in some form. Without something like this assumption there is no link between the data we observe and the data after a hypothetical intervention.
I think the kinds of examples that are drastically biased given Rubin’s “condition on everything” policy are not very common in practical data analysis problems, but it’s certainly easy to construct them. While I have not asked him, I suspect if I were to put a gun to Rubin’s head and gave him the above example, he will admit to not adjusting on W (and then say the situations in the example never happen in practice).
My view: M-bias is a special case of a more general issue where conditioning opens paths (due to how d-separation works in graphs). The way this issue manifests in practice is people assume they observe all confounders, adjust for them, get an estimate, and call it a day. In practice, their assumption is wrong, adjusting for all observable confounders opens a bunch of non-causal paths due to the inevitable presence of hidden variables, and the estimate they get is biased for this reason. There is, however, some evidence that this bias is sometimes not very big (I think Sander Greenland did some work on this)
Pearl likes graphs, but graphs are just a mathematical aid. What he and Rubin are talking about is not “about” graphs. You can prove all the theorems without graphs. Both Pearl and Rubin are talking about potential outcomes (interventionist view of causality). Pearl uses a model which makes cross-world independence assumptions (Rubin probably does not, although I have not asked him. Of course Rubin loves “principal stratification” which as far as I understand is wildly untestable, so who really knows what he thinks. A lot of workers in the field do not like cross-world independences because they are not testable).
To the extent that Rubin wants to estimate potential outcome random variables from observational data, he HAS to agree with Pearl on pain of bias (e.g. garbage). In the example I gave, if Rubin insists on conditioning on W, he will get a garbage answer for the potential outcome Y(x). Identification of potential outcomes isn’t the kind of thing where you can have a difference of opinion. It’s like having on opinion on what 2 + 2 is.
From your description, you say that Rubin insists on conditioning on all available data, so that includes W. But that doesn’t mean he has to get garbage, that just means he needs the right conditional.
Let Jaynes notation do the work. The base problem seems to be:
You can assign probabilities using observational data to create P(X1...XN | Intervention=No). How do I use that model to assign P(X1...XN | Intervention=Yes)?
Do these guys have any case where they make different predictions of what will happen in an intervention? Or do they just dance around in their own languages and come up with the same predictions?
“From your description, you say that Rubin insists on conditioning on all available data, so that includes W. But that doesn’t mean he has to get garbage, that just means he needs the right conditional.”
The right expression for p(y | do(x)) in this example should ignore W, that’s all there is to it. It’s not a notational issue.
“You can assign probabilities using observational data to create P(X1...XN | Intervention=No). How do I use that model to assign P(X1...XN | Intervention=Yes)?”
Good question! The answer is to use something called the consistency assumption (I think Pearl might call it “composition” in his book). This states, roughly that Y(X) = Y. (That is, observing Y when there is no intervention is the same as observing Y when X is intervened to attain whatever value it would naturally attain). This assumption is untestable, but to my knowledge every single paper in causal inference makes this assumption in some form. Without something like this assumption there is no link between the data we observe and the data after a hypothetical intervention.
I think the kinds of examples that are drastically biased given Rubin’s “condition on everything” policy are not very common in practical data analysis problems, but it’s certainly easy to construct them. While I have not asked him, I suspect if I were to put a gun to Rubin’s head and gave him the above example, he will admit to not adjusting on W (and then say the situations in the example never happen in practice).
My view: M-bias is a special case of a more general issue where conditioning opens paths (due to how d-separation works in graphs). The way this issue manifests in practice is people assume they observe all confounders, adjust for them, get an estimate, and call it a day. In practice, their assumption is wrong, adjusting for all observable confounders opens a bunch of non-causal paths due to the inevitable presence of hidden variables, and the estimate they get is biased for this reason. There is, however, some evidence that this bias is sometimes not very big (I think Sander Greenland did some work on this)