Learning Causal Direction from Repeated Observations over Time

Learning Causal Direction from Repeated Observations over Time Benjamin M. Rottman (benjamin.rottman@yale.edu) Frank C. Keil (frank.keil@yale.edu) Department of Psychology, Yale U., 2 Hillhouse Ave New Haven, CT 06520 Abstract Table 1: Example of How Order Can Influence Inferred Causal Direction Note. 1 stands for a positive mood and 0 stands for a negative mood. Inferring the direction of causal relationships is notoriously difficult. We propose a new strategy for learning causal direction when observing states of variables over time. When a cause changes state, its effects will likely change, but if an effect changes state due to an exogenous factor, its observed cause will likely stay the same. In two experiments, we found that people use this strategy to infer whether X→Y vs. X←Y, and X→Y→Z vs. X←Y→Z produced a set of data. We explore a rational Bayesian and a heuristic model to explain these results and discuss implications for causal learning. Day Keywords: causal reasoning; causal structures; time Introduction Learning the direction of a causal relationship from observation is notoriously difficult. Science students are taught that “correlation does not imply causation.” Indeed, the standard rational strategy proposed for how people learn causal structures (e.g., Steyvers et al., 2003) suggests that it is impossible to distinguish “Markov equivalent” structures such as [X→Y vs. X←Y] or [X→Y→Z vs. X←Y←Z vs. X←Y→Z]. In the three-variable structures above, all the three variables would be correlated. Additionally, X and Z would be conditionally independent given the state of Y. Because these structures have the same (in)dependence relations among variables, standard theories argue that it is impossible to learn which produced a set of data. One way that causal direction can be learned from observational data is if there is a temporal delay between the cause and effect (e.g., Lagnado & Sloman, 2006). If Linda gets sick on Monday and Sarah gets sick on Wednesday, it is likely that Linda gave Sarah the cold and not the reverse. However, temporal delay is not always available as a cue to causal direction. Sometimes temporal delays are too short to be perceptible. Or, the learner may only have access to periodic snapshots rather than a continuous stream of data. Here, we propose another way to learn the direction of causal relationships from observations over time. People may assume that when a cause changes its effects also change, but an effect may change due to an exogenous factor without its observed cause changing. This strategy is illustrated in the following example: Suppose you have two friends, Bill and Tim, and you are trying to figure out whether Bill’s mood influences Tim’s mood, or the reverse. For eight days, you observe whether Bill and Tim are in positive or negative moods. In Order 1 (Table 1), there are times when both Bill and Tim change between positive and negative moods simultaneously (e.g., Days 1-3), suggesting that there is some causal relationship between the two. Order 1: Bill→Tim Bill Tim Order 2: Bill←Tim Bill Tim Additionally, there are transitions in which Tim’s mood changes (e.g., Days 3-4) while Bill’s mood remains constant. These transitions suggest that some external event occurred to Tim (perhaps he did poorly on an exam), but the fact that Bill did not get into a bad mood on the same day suggests that Tim’s mood does not influence Bill’s mood. 1 Instead one might conclude that Bill’s mood influences Tim’s mood, which could explain why sometimes both of their moods change simultaneously. Critically, according to this account, the transitions from day to day are used to infer the causal direction. Consider Order 2, which has the exact same eight days as Order 1, except that Days 4 and 7 are switched. As in Order 1, there are times when both Bill and Tim’s moods change simultaneously (e.g., Days 1-3). However, Order 2 has transitions when Bill’s mood changes but Tim’s mood stays the same (e.g., Days 6-7; perhaps Bill got a job interview). This transition suggests that Bill’s mood does not influence Tim’s mood, so one might conclude that Tim’s mood influences Bill’s mood. All our experiments use manipulations like that in Table 1, in which one set of trials is rearranged in two different orders. If one ignores the temporal sequence of events it would be impossible to determine the causal direction; collapsing across the eight days, there is a correlation of .5, but correlation does not imply a causal direction. There are also some transitions when Tim’s mood changed and Bill’s mood stayed constant, but which might not suggest that Tim’s mood does not influence Bill’s mood. From Trials 4-5, Tim goes from a negative to positive mood, but Bill was already in a positive mood. Such transitions do not necessarily suggest that Tim’s mood does not influence Bill’s mood because Bill’s mood was already at “ceiling” or “floor.”