A Generative Model of Causal Cycles

A Generative Model of Causal Cycles Bob Rehder (bob.rehder@nyu.edu) Jay B. Martin (jbmartin@nyu.edu) Department of Psychology, New York University 6 Washington Place, New York, NY 10003 USA Abstract Unfolding Cycles Causal graphical models (CGMs) have become popular in nu- merous domains of psychological research for representing people’s causal knowledge. Unfortunately, however, the CGMs typically used in cognitive models prohibit representations of causal cycles. Building on work in machine learning, we pro- pose an extension of CGMs that allows cycles and apply that representation to one real-world reasoning task, namely, classi- fication. Our model’s predictions were assessed in experiments that tested both probabilistic and deterministic causal relations. The results were qualitatively consistent with the predictions of our model and inconsistent with those of an alternative model. We naturally reason about causally related events that occur in cycles. In economics, we expect that an increase in corpo- rate hiring may increase consumers’ income and thus their demand for products, leading to a further increase in hiring. In meteorology, we expect that melting tundra due to global warming may release the greenhouse gas methane, leading to yet further warming. In psychology, we expect that clin- icians will affect (hopefully help) their clients but also rec- ognize the clients often affect the clinicians. Many psychologists investigate causal reasoning using a formalism known as Bayesian networks or causal graphical models (hereafter, CGMs). CGMs are one hypothesis for how people reason with causal knowledge. There are claims that causal learning amounts to acquiring the structure and/or parameters of a CGM (Cheng, 1997; Gopnik et al., 2004; Griffiths & Tenenbaum, 2005; 2009; Lu et al., 2008; Sobel et al., 2004; Waldmann et al., 1995). And, many models of causal reasoning assume that people honor the inferential rules that accompany CGMs (Holyoak et al., 2010; Lee & Holyoak, 2008; Rehder & Burnett, 2005; Rehder, 2003; 2009; Rehder & Kim, 2010; Shafto et al., 2008; Sloman & Lagnado, 2005; Waldmann & Hagmeyer, 2005). Unfortunately, because standard CGMs prohibit the presence of causal cycles, these models are unable to repre- sent any of the cyclic events mentioned above. In this article, we take the initial steps to extend CGMs using an ‘unfolding’ trick from machine learning (Spirtes, 1993). We discuss the implications of this approach to one class of reasoning problem, namely classification. There is a rich literature on how causal knowledge among the features of a category changes how people classify. We first review evidence for causal cycles among category features and one proposal for how they affect classification. We then report two experiments that test that account. Finally, we present our own model for extending CGMs to represent cycles in people’s mental representations of categories. One technique used to elicit people’s beliefs about the cau- sal structure of categories is the theory drawing task. Sub- jects are presented with category features and asked to draw directed edges indicating how those features are causally related. These drawings show that causal cycles are com- mon. For example, Kim and Ahn (2002) found that 65% of subjects’ representations of mental disorders such as depres- sion included cycles. Sloman et al. found numerous cycles in subjects’ theories of everyday biological kinds and arti- facts. In a first attempt to account for how cycles affect cate- gorization, Kim et al. (2009) made two assumptions. The first was that causal knowledge affects classification in a manner specified by the dependency model (Sloman et al., 1998). On this account, features vary in their conceptual centrality, such that more central features provide more evi- dence for category membership. A feature’s centrality is a function of its number of (direct and indirect) dependents (i.e., effects). Quantitatively, feature i's centrality c i can be computed from the iterative equation, c i,t+1 = ∑d ij c j,t where c i,t is i's weight at iteration t and d ij is the strength of the causal link between i and its dependent j. For example, if a category has three features X, Y, and Z, and X causes Y which causes Z, then when c Z,1 is initialized to 1 and each causal link has a strength of 2, after two iterations the cen- tralities for X, Y, and Z are 4, 2, and 1. That is, feature X is more important to category membership than Y which is more important than Z. Qualitatively, the dependency model predicts this because X has two dependents (Y and Z), Y has one (Z), and Z has none. Kim et al.’s second assumption was that people reason with a simplified representation of cycles. Two reasons were provided for this assumption. First, because variables rarely cause each other constantly and simultaneously, it is likely that people assume that they influence each other in discrete time steps. Second, because it is implausible that people represent time steps extending into infinity, only a limited number of steps are likely to be considered. For example, consider the category in Fig. 1A in which feature C causes feature E and features X and Y are related in a causal cycle. Fig. 1B shows the cycle “unfolded” by one time step. The assumption is that in generation 1, X and Y mutually influ- enced one another, resulting in their states in generation 2 (X 2 and Y 2 ). Kim et al. proposed that feature importance would correspond to the predictions of the dependency model applied to the unfolded representation in Fig. 1B,