### General criterion for parameter identification

The parameter identification method described in Section 5.3.1 rests on two criteria: (1) The single door criterion of Theorem 5.3.1, and the back-door criterion of Theorem 5.3.2. This method may require appreciable bookkeeping in combining results from various segments of the graph. Is there a single graphical criterion of identification that unifies the two Theorems and thus avoids much of the bookkeeping involved?

This is a very insightful question, thanks!. The answer is: Yes. The unifying criterion is described in the following Lemma.

Lemma 1 (Graphical identification of direct effects)Let

cstand for the path coefficient assigned to the arrowX –> Yin a causal graphG. Parametercis identified if there exists a pair(W, Z), whereWis a single node inG(not excludingW=X), andZis a (possibly empty) set of nodes inG, such that:Zconsists of nondescendants ofY,Zd-separatesWfromYin the graphGformed by removing_{c}X–>YfromG.WandXared-connected, givenZ, inG._{c}Moreover, the estimand induced by the pair

(W, Z)is given by:The idea of unifying the two criteria came to me while reading Rod McDonald's recent paper "What can we learn from the path equations?" (Submitted). I have used this Lemma in a new report (R-276.pdf) / (R-276.ps), "Parameter identification: a new perspective". This report removes major inconsistencies in traditional definitions of parameter overidentification and offers graphical methods for deciding the degree to which a parameter is overidentified in a given model.

Best wishes,

========Judea Pearl

Comment by judea — February 21, 2007 @ 11:40 pm