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A transitivity heuristic of probabilistic causal reasoning

A Transitivity Heuristic of Probabilistic Causal Reasoning Momme von Sydow 1 (momme.von-sydow@bio.uni-goettingen.de) Bjorn Meder 1,2 (meder@mpib-berlin.mpg.de) York Hagmayer 1 (york.hagmayer@bio.uni-goettingen.de) Department of Psychology, University of Gottingen, Gosslerstr. 14, 37073 Gottingen, Germany Max Planck Institute for Human Development, Lentzeallee 94, 14195 Berlin, Germany Abstract In deterministic causal chains the relations „A causes B’ and „B causes C’ imply that „A causes C’. However, this is not necessarily the case for probabilistic causal relationships: A may probabilistically cause B, and B may probabilistically cause C, but A does not probabilistically cause C, but rather ¬C. The normal transitive inference is only valid when the Markov condition holds, a key feature of the Bayes net for- malism. However, it has been objected that the Markov as- sumption does not need to hold in the real world. In our stu- dies we examined how people reason about causal chains that do not obey the Markov condition. Three experiments involv- ing causal reasoning within causal chains provide evidence that transitive reasoning seems to hold psychologically, even when it is objectively not valid. Whereas related research has shown that learners assume the Markov condition in causal chains in the absence of contradictory data, we here demon- strate the use of this assumption for situations in which partic- ipants were directly confronted with evidence contradicting the Markov condition. The results suggest a causal transitivity heuristic resulting from chaining individual causal links into mental causal models that obey the Markov condition. Keywords: Transitivity; causal models; Markov condition; categorization, causal chain; syllogistic reasoning, heuristics Deterministic Causal Chains Deterministic causal relations imply transitivity: If A causes B, and B causes C, then A causes C. For deterministic rela- tions this can be justified on a purely logical basis. If one treats deterministic causal relations as material implications, the syllogism Modus Barbara applies, which is known since Aristotle. Expressed in terms of modern predicate logic it states that (x) (A(x) → B(x)) & (B(x) → C(x)) => (A(x) → C(x)). Also according to Mental Model Theory, transitivity is predicted (cf. Goodwin & Johnson-Laird, 2005). Like- wise, causal theories of reasoning, using causal strength estimates such as ΔP (Jenkins & Ward, 1965) or causal power (Cheng, 1997) entail transitivity for deterministic relationships. For instance, if P AB = P(B | A) – P(B | ¬A) = 1 and P BC = P(C | B) – P(C | ¬B) =1, it follows that P AC = 1. An analogous case can be made for causal power (Cheng, 1997) formalized by w = P AC / (1 – P(A | ¬C)). Probabilistic Causal Chains and the Markov Condition While the validity of transitive inferences in deterministic chains is undisputed, transitivity is not necessarily entailed when the causal relations are probabilistic. Nevertheless, transitivity may intuitively appear reasonable as well. For example, smoking (A) increases the probability of having tar (B) in your lungs which, in turn, is causally related to lung cancer (C). Thus, the three events constitute a generative causal chain ABC entailing that the probability of lung cancer is higher for smokers than for non-smokers. (i.e., P(C | A) > P(C | ¬A)). Thus, a transitive inference from A to C seems valid here as well. The representation of such causal relations in mental causal models (Sloman, 2005; Waldmann, 1996; Waldmann, Cheng, Hagmayer, & Blaisdell, 2008) as well as in Bayes nets (Spirtes, Glymour, & Scheines, 1993; Pearl, 2000) im- plies transitivity in causal chains. A Bayes net consists of nodes representing the domain variables and directed edges (“causal arrows”) representing the causal dependencies among the variables. At the heart of the Bayes nets formal- ism is the Markov condition, which states that a variable, conditioned on its direct causes, is independent of all other variables in the causal network except its effects (Hausman & Woodward, 1999; Pearl, 2000; Spirtes, Glymour, & Scheines, 1993). For example, applying the Markov condition to the causal chain ABC entails that A and C become in- dependent conditional on B. This assumption is important since it secures a modular representation of individual causal links, separate manipulability, and is crucial for infer- ring causal relations from probabilistic dependencies (Hausman & Woodward, 1999; Pearl, 2000). The Markov condition is also essential for basic inferences across complex causal networks because it allows for chain- ing the individual links to make quantitative predictions. For example, the conditional probability P(C | A) can be derived by a multiplicative combination of all possible paths leading from A to C: P(C | A) = P(B | A) P(C | B) + P(¬B | A) P(C | ¬B) (1) Thus, the Markov condition allows us to infer the con- ditional probability P(C | A) by combining the causal links constituting the chain without observing this relation di- rectly. The Markov assumption has been postulated to be a nec- essary and universal feature of causal relations in the world (Hausman & Woodward, 1999). However, the Markov con- dition has also been criticized. Particularly Cartwright (2001, 2002) argued that the proof for the necessity of the Markov condition in the deterministic case is valid, but va- cuous, and the proof of necessity in the probabilistic case is invalid or at least question begging. Markov Condition on the Type Level It is possible that some data may provide evidence for a (probabilistic) generative causal relation between events A and B on the one hand and events B and C on the other hand, but this does not necessarily entail that A and C are