TETRAD and SEM.

The authors have given us a lucid, concise, and accurate account of the TETRAD project, an account likely to clear much of the confusion over the philosophy of graph-based causal discovery methods. As one who has been active in the exploration and formulation of this philosophy, I am very pleased to see TETRAD arrive at this stage of development and now presented as a practical modeling tool for the research community. In this commentary, I wish to expand an aspect of TETRAD which the authors, by using the analogy with ordinary statistical estimation, have played down: the importance of TETRAD to the foundations of structural equations modeling (SEM). The assumptions underlying statistical estimation are of fundamentally diierent character from those underlying causal modeling tasks, and TETRAD, by fostering an understanding of these diierences, can help establish not only its own legitimacy but the legitimacy of the entire SEM enterprise. Thus, I will rst add a few arguments in support of the assumptions underlying TETRAD and then attempt to show that, even if these assumptions are not fully embraced, the general lessons learned from graphical causal modeling are destined to have a profound impact on SEM's practice and philosophy. The operation of TETRAD is based on two assumptions: \causal independence" and \faithfulness." 1 While the assumption of causal independence is rarely challenged, the faithfulness assumption has drawn quite a few objections and is a likely target for further criticism from SEM researchers. I would like therefore to elaborate some additional rationale for faithfulness. Because TETRAD relies primarily on nonexperimental data, causal claims are issued with guarantees weaker than those obtained through controlled randomized experiments. Pearl and Verma (1991) expressed these guarantees in terms of two orthogonal notions: minimality and stability. 2 Minimality guarantees that any alternative structure compatible 1 Causal independence, also known as \Reichenbach's Principle" or \no correlation without causation," is rooted in the principle of no action at a distance Arntzenius, 1990]. The notion of faithfulness (also called \DAG-isomorphism" and \nondegeneracy" Pearl, 1988, p. 391]) was termed \stability" by Pearl and Verma (1991) to emphasize the invariance of certain independencies to functional form. 2 These guarantees were advanced in connection with the discovery algorithm IC (for \Inferred Causation") which was developed in parallel with and on the same principles as the PC algorithm of Spirtes et al. (1993).