Discovering hidden causes using statistical evidence
Discovering hidden causes using statistical evidence Christopher G. Lucas Kenneth Holstein Charles Kemp c.lucas@ed.ac.uk School of Informatics University of Edinburgh kjh64@pitt.edu Department of Psychology University of Pittsburgh ckemp@cmu.edu Department of Psychology Carnegie Mellon University Abstract ple might solve this problem, including a model that focuses on noisy OR and noisy AND NOT relationships. We then present an experiment that allows us to compare the models and reveals patterns of judgments that standard accounts of causal reasoning do not anticipate. People frequently reason about causal relationships and vari ables that cannot be directly observed. This paper presents results from an experiment in which participants used statistical information to make judgments about the number and base rates of hidden causes, as well as the forms of causal relationships in which those causes participated. Our data allow us to evalu ate several models of hidden cause discovery, and reveal that people have different expectations about the forms of causal relationships than recent theories predict. Keywords: causal inference; hidden causes; minimum descrip tion length; functional form A simple causal discovery problem This paper will focus on inferences about a causal system that has a single visible cause v and a single effect e. In addition to the visible cause, there may be one or more hidden causes h i that influence the effect. For simplicity, we assume that all variables are binary. Suppose that the system is observed over many trials in each case the visible cause v takes value 0 or 1, and the effect e takes value 0 or 1. The resulting sequence of observations can be summarized by a pair of statistics [p 0 p] [P(e 1|v 0) P(e 1|v 1) that indicates the probability that e 1 when v is either 0 or 1. For our purposes, the space of all possible pairs [p 0 p] can be organized into a handful of qualitatively different classes. Figure 1a shows representatives of the five classes that we con sider in this paper. The classes can be enumerated by taking three criteria into account. First, probabilistic relationships between v and e provide evidence of hidden causes, and we therefore distinguish between extreme statistics that equal 0 or 1 and intermediate statistics that fall between 0 and 1. For example, pairs [0, 1] and [p, 1] in Figure 1a are qualitatively different because [0, 1] has two extreme statistics and [p, 1] has only one extreme statistic. The second and third criteria are based on non generic relationships that may hold between p and p 0 The second criterion requires that we distinguish be tween pairs for which p p 0 and pairs for which the statistics are not equal. The third criterion requires that we distinguish between pairs for which p 1 p 0 and pairs that do not have this property. For example, pairs [p, p] and [p 1 p 2 in Figure 1a can be distinguished using the second criterion, and pairs [p, 1 p] and [p 1 p 2 can be distinguished using the third crite rion. Applying the three criteria and collapsing across all other differences between pairs produces a total of six equivalence classes. The class not shown in Figure 1a includes the pairs [1, 1] and [0, 0] and we dropped this class because the effect variable is constant. Although the relationship between v and e may appear to be probabilistic, we assume that e is a deterministic function of v and potentially one or more hidden variables. This assump tion of causal determinism is consistent with several previous formal approaches (Pearl, 2000; Lucas Kemp, 2012) and with previous suggestions that people are causal determin ists (Schulz Sommerville, 2006; Frosch Johnson Laird, In order to reason fluently about cause and effect, humans must frequently make inferences about causes that are hid den from direct observation. For example, we explain the behaviors of agents by appealing to mental states, we infer the presence of illnesses from symptoms, and we can diagnose hidden faults in mechanical and electrical systems by seeing how they fail. This ability is essential for explanation, predic tion, and planning, but exactly how we reason about hidden causes, or even come to believe that hidden causes are present in the first place, is not yet well understood. What are the signs that a hidden cause might be present in a causal system? One is unreliability: hidden factors are likely to be at work when a relationship appears to be un reliable or stochastic, and there is no observable reason for this behavior. Previous work has explored how and when people posit hidden causes to explain unreliable relationships. When children observe a failure prone electromechanical de vice, they tend to infer that its failures are due to a hidden controller (Schulz Sommerville, 2006) People can also in fer common hidden causes when unreliable effects tend to be correlated (Kushnir, Gopnik, Lucas, Schulz, 2010; Carroll Cheng, 2009; Mayrhofer, Goodman, Waldmann, Tenen baum, 2008) and can make inferences about the nature of hidden causes from their interactions with background rates of effects (Carroll Cheng, 2010) Despite these results, there are many open questions about how humans use statistical information to discover hidden causes. In particular, there are no comprehensive studies that explore how people infer the form of the function that specifies how observed and hidden causes combine to produce an effect. Many theories assume that prospective causes combine ac cording to noisy OR and noisy AND NOT functions, but we introduce an experimental paradigm that reveals that people naturally entertain a broader range of causal functions. The next section introduces the simple problem of causal discovery that we consider. The following section describes three models that capture different theories about how peo
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