The Causal Markov Condition: Should you choose to accept it? Karen R. Zwier Department of History and Philosophy of Science University of Pittsburgh

The Causal Markov Condition SGS [1993, 2000] Formulation: Let G be a causal graph with vertex set V and P be a probability distribution over the vertices in V generated by the causal structure represented by G. G and P satisfy the Causal Markov Condition if and only if for every W in V, W is independent of V \ (Descendants(W)  Parents(W)) given Parents(W). The debate over the Causal Markov Condition (CMC) has largely taken place at the logical/metaphysical level From the definition above, it should be obvious that this relation won’t hold between arbitrary G and P. Therefore, criticisms that pick out “counterexamples”—pairs of G and P for which the CMC does not hold, are not actually criticisms of the CMC. These are criticisms of naïve use of the CMC. And they make known to us interesting situations in which statistical modeling decisions affect the applicability of the CMC. Introduction Critique of Metaphysical Approach CMC Breakdown Pragmatic Approach Conclusion

Interesting results of “counterexample” criticisms: Cyclical graphs Causal insufficiency Logical relationships among variables Selection bias / Sampling bias Inter-Unit Causation Non-homogeneous populations Introduction Critique of Metaphysical Approach CMC Breakdown Pragmatic Approach Conclusion

Where we’re going… There is another type of criticism against the CMC: what I call metaphysical criticism. The debate over the Causal Markov Condition (CMC) has largely taken place at the logical/metaphysical level. My claim: The validity of the CMC cannot be decided on a metaphysical basis My alternative: pragmatic, material considerations should decide use/non-use of the CMC Introduction Critique of Metaphysical Approach CMC Breakdown Pragmatic Approach Conclusion

Hume’s problem Causation is a non-logical, non-conceptual dependence. Therefore, there is nothing in the concepts of the related objects that tells us that one causes the other. Only objects are observable; causation is not. Even if we allow that causation, or a “causal power” was operative in a certain situation, we still cannot extend this assumption to future instances because of the general problem of induction. Introduction Critique of Metaphysical Approach CMC Breakdown Pragmatic Approach Conclusion

Hume’s problem gets worse 1.Hume did not consider concepts to be problematic. For Hume, sense data automatically turns into an idea. –But concepts are problematic, especially in a scientific discussion of causation, where our everyday notions and sense data may not map on to the entities of our theories. The decision of which variables to consider is not trivial. And there are many other non-trivial modeling decisions as well. 2.For Hume, “necessary connection” is essential to causation. –But in our framework, causation is not limited to necessary connection. We want to accommodate a probabilistic notion of causation as well. But what is the connection between probability and causation? This is what is under debate in the CMC. Introduction Critique of Metaphysical Approach CMC Breakdown Pragmatic Approach Conclusion

So now what? Hume’s argument does not prove that causation is not real (i.e. is only an artifact of our minds). It only proves that we can’t be certain that it is real. So if, even in the face of Hume’s argument, we choose to be realists about causation (and I do!), we still must learn from Hume and take our epistemic limitations seriously. Specifically, because of all of the diverse modeling possibilities I showed in the last slide, we cannot make we cannot make inference decisions (i.e. assumptions about the connection between probability and causation) on a metaphysical basis. We must make these decisions on a pragmatic basis, using the material considerations of the situation at hand, after we have already made data collection decisions. −Data collection decisions: What units? What variables? What possible values for those variables? What population? How to sample? −Inference decisions: How do we go from our data to a causal hypothesis? Specifically, what connection should we assume between causation and probability? Introduction Critique of Metaphysical Approach CMC Breakdown Pragmatic Approach Conclusion

The Causal Markov Condition …is a potential assumption about the connection between causation and probability: specifically, an assumption about the relationship between a causal graph and the probability distribution over its variables. SGS [1993, 2000] Formulation: Let G be a causal graph with vertex set V and P be a probability distribution over the vertices in V generated by the causal structure represented by G. G and P satisfy the Causal Markov Condition if and only if for every W in V, W is independent of V \ (Descendants(W)  Parents(W)) given Parents(W). Introduction Critique of Metaphysical Approach CMC Breakdown Pragmatic Approach Conclusion

Breaking down the CMC The vertex set V \ (Descendants(W)  Parents(W)) can be partitioned into the following vertex sets: 1.NPA(W): All non-parental ancestors of W that are not also in Descendants(W); 2.Siblings(W): All siblings of W that are not also in Descendants(W)  Parents(W); 3.Co-Ancestors(W): All ancestors A of any vertex D in Descendants(W), where A is not in Descendants(W)  Ancestors(W)  Siblings(W); 4.UnrelatedExogenous(W): All exogenous vertices in the graph that are not also in Ancestors(W)  Co-Ancestors(W); and 5.OtherDescendants(W): All descendants D of any vertex in NPA(W)  Siblings(W)  Co-Ancestors(W)  UnrelatedExogenous(W), where D is not in Descendants(W)  Ancestors(W)  Siblings(W)  Co- Ancestors(W). Introduction Critique of Metaphysical Approach CMC Breakdown Pragmatic Approach Conclusion

Breaking down the CMC W V \ (Descendants(W)  Parents(W)) | Parents(W) entails that: 1.WNPA(W) | Parents(W) 2.WSiblings(W) | Parents(W) 3.WCo-Ancestors(W) | Parents(W) 4.W UnrelatedExogenous(W) | Parents(W) 5.W OtherDescendants(W) | Parents(W)       Introduction Critique of Metaphysical Approach CMC Breakdown Pragmatic Approach Conclusion

So what would a pragmatic decision to use/not use the CMC look like? On the basis of the modeling decisions we have made in the data-gathering phase (e.g. units, variables, etc.) we may or may not want to assume all of the conditional independence statements made by the CMC. We can decide to assume a subset of these! Introduction Critique of Metaphysical Approach Pragmatic Approach Specifics ConclusionIntroduction Critique of Metaphysical Approach CMC Breakdown Pragmatic Approach Conclusion

W NPA(W) | Parents(W) This assumption is called robustness. The concept of robustness comes from a common way of understanding physical causation, in which the set of circumstances immediately preceding an effect is enough to determine that effect. Robustness between a variable A and another variable B means that B is unaffected by small disturbances in how A comes about Given the parents (i.e. direct causes) of a variable W, the non- parental ancestors (NPA(W)) have no influence whatsoever on the value of W. Only the direct causes of a vertex W in the graph have a “special” causal power over W. Introduction Critique of Metaphysical Approach CMC Breakdown Pragmatic Approach Conclusion 

Keep/Drop Robustness? Robustness is a standard that, although desirable for reductive physical accounts, can be difficult to satisfy: it says that for every variable W in V, we have a complete set of direct causes that screens off all other ancestors. But sometimes this assumption is not necessary for our purposes… Introduction Critique of Metaphysical Approach Pragmatic Approach Specifics ConclusionIntroduction Critique of Metaphysical Approach CMC Breakdown Pragmatic Approach Conclusion

Example I am buying a tennis racquet. In order to inform my choice, I would like to know something about the causal relationship between the price of a tennis racquet (P) is a cause of tennis- playing success (S). Setting 1: My goal is simply to find out if P is a cause of S, so I can better my tennis playing. I may consider other variables as well, but I have no desire to fine-tune my causal model—to find out if P is a necessary member of a set of direct causes of S, or a necessary member of a set of direct causes of one of the ancestors of S.  Here, do not assume robustness in inferring causal graph. Setting 2: My goal is to maximize the success of my tennis playing while expending as little effort as possible to “intervene” on my condition. I want to know about the precise relationship between P and S within a network of other variables, so that if another set of variables screens off P, I will no longer worry about the price of my tennis racquet.  Here, assume robustness in inferring causal graph. Introduction Critique of Metaphysical Approach Pragmatic Approach Specifics ConclusionIntroduction Critique of Metaphysical Approach CMC Breakdown Pragmatic Approach Conclusion

W Siblings(W) | Parents(W) This assumption is Reichenbach’s Principle of the Common Cause: “…the common cause is the connecting link which transforms an independence into a dependence.” One goal we often have in science is to separate phenomena into independent realms so that we can study them more accurately. Introduction Critique of Metaphysical Approach CMC Breakdown Pragmatic Approach Conclusion 

Keep/Drop PCC? Example. The Principle of the Common Cause is controversial particularly in the context of EPR correlations. Setting 1: We mean to emphasize that entangled particles are not independent of each other (and in fact, they are perfectly anti- correlated). Here, we might choose to represent the measured spin of each of the particles with a separate variable and discard the principle of the common cause, allowing a correlation to exist between the effects.  Do not assume PCC when inferring the causal graph. Setting 2: We mean to emphasize the separable variables of the system. Since the two entangled particles are never separable in their recorded measurements (as far as we know), we might choose to represent the two measurements together in one variable.  Here, we assume the PCC when inferring the causal graph. Introduction Critique of Metaphysical Approach Pragmatic Approach Specifics ConclusionIntroduction Critique of Metaphysical Approach CMC Breakdown Pragmatic Approach Conclusion

Conclusions Metaphysical debate over the CMC gets us nowhere, because we don’t have the necessary epistemic access to the nature of causation We can break the CMC down into its component conditional independence statements and pick and choose from them in a given situation Note: A weaker assumption means that the underdetermination problem is worse—the hypothesis space is increased. But there is a trade-off: an assumption that is too strong for our purpose may eliminate the very hypothesis that we want to consider. A job for the future: formulating the algorithms based on weakened CMC assumptions Introduction Critique of Metaphysical Approach CMC Breakdown Pragmatic Approach Conclusion