Independence and Conditional Independence in Causal Systems

This paper studies the interrelations between independence or conditional independence and causal relations, defined in terms of functional dependence, that hold among variables of interest within the settable system framework of White and Chalak. We provide formal conditions ensuring the validity of Reichenbach’s principle of common cause and introduce a new conditional counterpart, the conditional Reichenbach principle of common cause. We then provide necessary and sufficient causal conditions for probabilistic dependence and conditional dependence among certain random vectors in settable systems. Immediate corollaries of these results constitute causal conditions sufficient to ensure independence or conditional independence among random vectors in settable systems. We demonstrate how our results relate to and generalize results in the artificial intelligence literature, particularly results concerning d-separation. PRELIMINARY AND INCOMPLETE Acknowledgment: The authors thank Julian Betts, Graham Elliott, Clive Granger, Mark Machina, Dimitris Politis, and especially Ruth Williams for their helpful comments and suggestions. All errors and omissions are the authors’ responsibility. 1 Karim Chalak, Department of Economics, Boston College, 140 Commonwealth Avenue, Chestnut Hill, MA 02467 (email: [email protected]). Halbert White, Department of Economics 0508, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0508 (email: [email protected]).