ARTICLE Communicated by Terrence Sejnowski Causality, Conditional Independence, and Graphical Separation in Settable Systems Karim Chalak Downloaded from http://direct.mit.edu/neco/article-pdf/24/7/1611/1065694/neco_a_00295.pdf by guest on 26 September 2021 [email protected] Department of Economics, Boston College, Chestnut Hill, MA 02467, U.S.A. Halbert White [email protected] Department of Economics, University of California, San Diego, La Jolla, CA 92093, U.S.A. We study the connections between causal relations and conditional in- dependence within the settable systems extension of the Pearl causal model (PCM). Our analysis clearly distinguishes between causal notions and probabilistic notions, and it does not formally rely on graphical representations. As a foundation, we provide definitions in terms of suit- able functional dependence for direct causality and for indirect and total causality via and exclusive of a set of variables. Based on these founda- tions, we provide causal and stochastic conditions formally characterizing conditional dependence among random vectors of interest in structural systems by stating and proving the conditional Reichenbach principle of common cause, obtaining the classical Reichenbach principle as a corol- lary. We apply the conditional Reichenbach principle to show that the useful tools of d-separation and D-separation can be employed to estab- lish conditional independence within suitably restricted settable systems analogous to Markovian PCMs. 1 Introduction This article studies the connections between probabilistic conditional inde- pendence and notions of causality based on functional dependence. Our results shed light on two questions fundamental to the understanding of empirical relationships. First, what implications for the joint probability distribution of variables of interest derive from knowledge of functionally defined causal relationships between them? Conversely, what restrictions (if any) on the possible causal relationships holding between variables of in- terest follow from knowledge of the joint probability distribution governing these variables? These questions lie at the heart of Reichenbach’s (1956) principle of com- mon cause, which holds that if two random variables are correlated, then Neural Computation 24, 1611–1668 (2012) c 2012 Massachusetts Institute of Technology 1612 K. Chalak and H. White one causes the other or there is an underlying common cause. They are also addressed in the context of the Pearl causal model (PCM) (Pearl, 2000) by such notions as d-separation and D-separation (Geiger, Verma, & Pearl, 1990). Nevertheless, the status of Reichenbach’s venerable principle is still ambiguous (Dawid, 2010a), and results for d-separation and D-separation establish conditional independence under a “Markovian” condition that re- quires the existence of certain jointly independent “background variables.” Not only is the Markovian condition strong, but it intermingles functionally Downloaded from http://direct.mit.edu/neco/article-pdf/24/7/1611/1065694/neco_a_00295.pdf by guest on 26 September 2021 determined causal notions with probabilistic conditions (independence) to deliver its results. The main goal of this article, then, is to answer the two questions we have set out by connecting causal concepts founded on functional dependence to notions of probabilistic dependence without imposing strong Markovian- type assumptions and in such a way that the roles of causality and prob- ability are clearly delineated. Specifically, our main results provide new necessary and sufficient conditions for conditional independence relations to hold among structurally related variables of interest. These results are not accessible using the PCM. This delivers a generally applicable framework for disentangling the causal relations underlying probabilistic dependence, thereby facilitating learning about empirical relations in both experimental and nonexperimental contexts. (See Chalak & White, 2011, for a taxonomy of the use of conditional independence relations in this context.) We obtain our results using the settable systems (SS) causal framework of White and Chalak (2009; referred to hereafter as WC), an extension of the PCM. As WC and White, Chalak, and Lu (2010) discuss and illustrate with numerous examples, SS can better accommodate systems involving optimization, nonunique equilibrium, and learning. Among the features distinguishing SS from the PCM are SS’s lack of a unique fixed point re- quirement for the structural equations, and its “unlumping” of the PCM’s background variables into fixed system attributes that do not act causally and structurally exogenous variables that can play a causal role. To intro- duce SS here and to concretely illustrate its advantages for our purposes relative to the PCM or the causal framework of Spirtes, Glymour, and Scheines (1993; referred to hereafter as SGS), we use a simple advice-action- outcome example in which an expert (e.g., a physician or financial advisor) advises an agent (e.g., a patient or investor) who is undertaking an action that influences an outcome of interest. In pursuing our goal, we make a number of related further contributions. First, by operating within SS, our results connecting causal and probabilistic relations are valid in contexts where other causal frameworks, such as PCM or SGS, may not apply. Second, to provide a foundation for connecting causal and probabilistic dependence, we introduce new rigorous and general function-based defi- nitions of direct causality, as well as of indirect and total causality via and exclusive of a set of variables. We use our advice-action-outcome example Causality and Conditional Independence in Settable Systems 1613 to demonstrate the intuitive content of our definitions. Although these def- initions may lend themselves to convenient graphical representations, they do not rely on properties of graphs. Our notions extend and complement definitions for indirect and path-specific effects in Pearl (2001) and Avin, Shpitser, and Pearl (2005) and related notions of direct, indirect, and total effects proposed in Robins and Greenland (1992), SGS, Robins (2003), Rubin (2004), Didelez, Dawid, and Geneletti (2006), and Geneletti (2007). Third, using these foundations, we establish general connections be- Downloaded from http://direct.mit.edu/neco/article-pdf/24/7/1611/1065694/neco_a_00295.pdf by guest on 26 September 2021 tween causal and probabilistic dependence by stating and proving a new result, the conditional Reichenbach principle of common cause. As a corol- lary,this makes rigorous the classical (unconditional) Reichenbach principle of common cause. Among other things, this clarifies ambiguities in the lit- erature surrounding the formal status of Reichenbach’s principle (Spohn, 1980; Hausman & Woodward, 1999; Cartwright, 2000; Dawid, 2010a). Sig- nificantly, the conditional Reichenbach principle holds for any probability measure and does not require Markovian structure, as in Geiger et al. (1990), SGS, or Pearl (1993, 1995, 2000). Because the conditional Reichenbach prin- ciple is not restricted to Markovian systems, it permits a clear separation between causal relations and probabilistic relations. Fourth, we apply SS and the conditional Reichenbach principle to shed light on a variety of results from the PCM and DAG literature connecting causality and conditional independence. We show that the useful tools of d-separation and D-separation (Geiger et al., 1990) can be applied within a suitably restricted SS, analogous to Markovian PCMs. Specifically, we show that the conditional Reichenbach principle ensures the directed lo- cal Markov property (Lauritzen, Dawid, Larsen, & Leimer, 1990) among variables generated by the unimpeded evolution of the restricted settable system. In such systems, d-separation and D-separation (Geiger et al., 1990) can be used to establish conditional independence. We show as well how causal intuitions associated with these graphical separation principles can fail, even in Markovian systems, or hold, even in non-Markovian sys- tems. Correspondingly, the PCM and DAG literature recognizes that for Markovian PCMs, d-separation is sufficient but not necessary for condi- tional independence. To accommodate this, SGS refer to distributions in which failure of d-separation implies conditional dependence as “faithful” and Pearl (2000) refers to such distributions as “stable.” It is not clear, how- ever, what is excluded by faithfulness and stability restrictions. The need for these qualifications arises in part because the graphical PCM and DAG causal semantics is not sufficiently rich to accommodate functional defini- tions of causality. Because the conditional Reichenbach principle does not rely on graphs, and therefore does not rely on d-separation, it offers deeper insight into the notions of faithfulness and stability. The PCM and DAG lit- erature also establishes that conditioning on “common successors” induces conditional dependence among “causes” in faithful or stable Markovian systems (SGS; Pearl, 2000). We provide mild restrictions on probability 1614 K. Chalak and H. White measures and response functions demonstrating that this property holds not only for faithful or stable Markovian systems but also systems that need not be faithful or stable or Markovian. Taken together, the results of this article show how the SS extension of the PCM overcomes several cogent criticisms of the use of PCM DAGs for the study of the connections between causality and conditional inde-