# Causal Analysis in Theory and Practice

## February 27, 2007

### Counterfactuals in linear systems

Filed under: Counterfactual,Linear Systems — judea @ 4:08 pm

What do we know about counterfactuals in linear models?

Here is a neat result concerning the testability of counterfactuals in linear systems.
We know that counterfactual queries of the form P(Yx=y|e) may or may not be empirically identifiable, even in experimental studies. For example, the probability of causation, P(Yx=y|x',y') is in general not identifiable from experimental data (Causality, p. 290, Corollary 9.2.12) when X and Y are binary.1 (Footnote-1: A complete graphical criterion for distinguishing testable from nontestable counterfactuals is given in Shpitser and Pearl (2007, upcoming)).

This note shows that things are much friendlier in linear analysis:

Claim A. Any counterfactual query of the form E(Yx |e) is empirically identifiable in linear causal models, with e an arbitrary evidence.

Claim B. E(Yx|e) is given by

E(Yx|e) = E(Y|e) + T [xE(X|e)]      (1)

where T is the total effect coefficient of X on Y, i.e.,

T = d E[Yx]/dx = E(Y|do(x+1)) – E(Y|do(x))      (2)

Thus, whenever the causal effect T is identified, E(Yx|e) = is identified as well.

## February 22, 2007

### Moderatorâ€™s Update

Filed under: Announcement — moderator @ 10:20 pm

Thanks for visiting! We have been overwhelmed by the interest in our new blog, and we hope to continue improving this forum with your suggestions and comments (please keep them coming!) To our new visitors: we welcome you to browse through our archives and invite you to contribute any questions or opinions that you may have on the topic. Please keep in mind that while previous posts have been taken from discussions regarding the book "Causality" by Judea Pearl, the scope of this blog is not intended to be limited to a particular book or view. As we continue to develop the blog, you can expect to see new content including regular commentaries and topics of interest, conference/workshop announcements, abstracts/reviews of recent articles, and most importantly, your contributions. We hope to hear from you!

### Back-door criterion and epidemiology

Filed under: Back-door criterion,Book (J Pearl),Epidemiology — moderator @ 9:03 am

The definition of the back-door condition (Causality, page 79, Definition 3.3.1) seems to be contrived. The exclusion of descendants of X (Condition (i)) seems to be introduced as an after fact, just because we get into trouble if we dont. Why cant we get it from first principles; first define sufficiency of Z in terms of the goal of removing bias and, then, show that, to achieve this goal, you neither want nor need descendants of X in Z.