Learning Causal Direction from Transitions with Continuous and Noisy Variables

Learning Causal Direction from Transitions with Continuous and Noisy Variables Kevin W. Soo (kws10@pitt.edu) Benjamin M. Rottman (rottman@pitt.edu) Department of Psychology, University of Pittsburgh 3939 O’Hara Street, Pittsburgh, PA 15260 Abstract two friends, Friend X and Friend Y over time, and to try to infer whether Friend X’s mood influences Friend Y’s mood, or vice versa. The X and Y columns in Table 1 show the kind of data presented to participants in their experiments. Each person could either be in a positive (1) or negative (0) mood. In order to explain this reasoning process, it is easiest to first think about how a particular causal structure tends to produce certain types of transitions given “shocks” to X or Y. A shock is an economic term that means an exogenous event that produces a change in an observed variable. Previous work has found that one way people infer the direction of causal relationships involves identifying an asymmetry in how causes and effects change over time. In the current research we test the generalizability of this reasoning strategy in more complex environments involving ordinal and continuous variables and with noise. Participants were still able to use the strategy with ordinal and continuous variables. However, when noise made it difficult to identify the asymmetry participants were no longer able to infer the causal direction. Keywords: causal reasoning; causal structure; time Table 1: Transition-based Learning with Binary Variables. “S(X)” means a shock to X. Introduction Knowing the direction of causal relations is critical for an agent to be able to act effectively in the world. For example, knowing that James gossips to Julie but not vice versa would allow one to selectively share a secret. Knowing that energy prices influence the price of produce (due to shipping costs) but not vice versa would allow one to predict the prices of energy and produce after an oil embargo or after a poor harvest. There are four known strategies that lay people use to infer causal direction between two variables, X and Y. The first strategy is intervention. Imagine that one intervenes and sets the value of X to either 1 or 0. If the distribution of Y is different when X is set to 1 vs. 0, it implies that X causes Y (e.g. Steyvers et al., 2003) The remaining three strategies apply to “observational” data, when a learner observes the states of X and Y. Inferring causal direction from observational data is notoriously tricky, and these strategies are usually viewed as heuristics. The next strategy is causal sufficiency. In some situations learners have a strong belief that when a cause is present its effect would also be present. In this case, observing [x = 0, y = 1] is consistent with inferring X→Y but not X←Y (Mayrhofer & Waldmann, 2011; cf. Deverett & Kemp, 2012). Another strategy is to reason using temporal “delay” or “order”. If one observes that X usually occurs before Y (e.g., X changes from 0 to 1, and subsequently Y changes from 0 to 1, or Event X occurs and then Event Y occurs) people infer that X→Y instead of X←Y (e.g. Lagnado & Sloman, 2006). Learning Causal Direction from Transitions Rottman & Keil (2012) proposed that people also learn causal direction in a fourth way, by reasoning about changes or “transitions” in variables over time. For example, Rottman & Keil asked participants to observe the moods of Time X Y Tran- sition Type Is transition consistent with causal structure? X→Y X←Y ΔXΔY Yes: S(X) Yes: S(Y) ΔY Yes: S(Y) No Δ Yes: No S No: S(Y) would change X Yes: No S Assume that X→Y is the true causal structure. When something produces a change in X’s mood (a shock to X), the change in X’s mood will generally produce a change in Y’s mood. Transitions in which both X and Y change are labeled as “ΔXΔY”. Now consider what would happen given a shock that changes Person Y’s mood. If there is reason to believe that the states of X and Y are “temporally dependent” or autocorrelated, that people’s moods are fairly stable over time, then when there is a change in Y’s mood X’s mood would usually remain stable (i.e. in the same state as it was before). Transitions in which there is a change in Y but no change in X are labeled as “ΔY”. Finally, transitions in which both X and Y stay the same are labeled “No Δ” for no change. No Δ transitions are common under both X→Y and X←Y assuming that X and Y are autocorrelated. This logic implies that if X→Y is the true causal direction, then ΔXΔY and ΔY transitions would be common; however, ΔX transitions would be rare because if X’s mood changes then there would typically be a change in Y as well (ΔXΔY instead of ΔX). Alternatively, if the true causal structure is X←Y, then ΔXΔY and ΔX would be common, but ΔY would be rare. Given this logic, one can reason backwards to infer the direction of the causal relation. For example, in Table 1, there is a ΔXΔY and a ΔY transition but no ΔX transitions, implying that X→Y is more likely to be the true causal structure than X←Y. The