Running Head : BAYES AND BLICKETS Bayes and blickets : Effects of knowledge on causal induction in children and adults

People are adept at inferring novel causal relations, even from only a few observations. Prior knowledge about the probability of encountering causal relations of various types and the nature of the mechanisms relating causes and effects plays a crucial role in these inferences. We test a formal account of how this knowledge can be used and acquired, based on analyzing causal induction as Bayesian inference. Five studies explored the predictions of this account with adults and 4-year-olds, using tasks in which participants learned about the causal properties of a set of objects. The studies varied the two factors that our Bayesian approach predicted should be relevant to causal induction: the prior probability with which causal relations exist, and the assumption of a deterministic or a probabilistic relation between cause and effect. Adults’ judgments (Experiments 1, 2, and 4) were in close correspondence with the quantitative predictions of the model, and children’s judgments (Experiments 3 and 5) agreed qualitatively with this account. Bayesian inference and causal learning 3 Bayes and blickets: Effects of knowledge on causal induction in children and adults As adults, we know a remarkable amount about the causal structure of our environment. Discovering this structure is a difficult inductive problem, requiring unobservable causal relations to be inferred from limited observed data. Historically, psychological theories of causal induction have fallen into two camps (Newsome, 2003): covariation-based approaches characterize human causal induction as the consequence of a domain-general statistical sensitivity to covariation between cause and effect (e.g., Cheng, 1997; Shanks, 1995), while mechanism-based approaches focus on the role of prior knowledge about causal mechanisms (e.g., Ahn & Kalish, 2000; Bullock, Gelman, & Baillaergon, 1982; Shultz, 1982; Wolff, 2007). In this paper, we argue that a central part of the explanation for how people come to know so much about the causal structure of their world is that they are capable of combining these sources of information, using domaingeneral statistical inference guided by domain-specific prior knowledge. We show how covariational evidence and prior knowledge about causal mechanisms can be combined by a mechanism for causal reasoning based on Bayesian inference. We test the predictions of the resulting formal account through a series of experiments with both adults and children. Bayesian inference provides a natural way to identify how covariational evidence and prior knowledge should be combined, indicating how a rational learner could best arrive at an accurate solution to the problem of inferring causal structure from observed data. The resulting computational-level analysis (in the spirit of Marr, 1982) of the problem of causal induction is analogous to work in ideal observer or signal detection theory (Green & Swets, 1966; Yuille & Kersten, 2006), which indicates how a visual system can best make inferences Bayesian inference and causal learning 4 about the world from visual data. Just as ideal observer models make it possible to explore how statistical information about the kinds of things encountered in the world guides perception, Bayesian inference about causal structure gives us a way to investigate how statistical information about events co-occurring interacts with existing knowledge to guide human causal learning. In order to provide a simple, concrete setting in which to explore the interplay of covariational evidence and prior knowledge, we develop our approach for the specific case of learning about the causal relations between objects in simple physical systems. We focus on the blicket detector paradigm (Gopnik & Sobel, 2000; Gopnik, Sobel, Schulz, & Glymour, 2001; Sobel et al., 2004): Adults or children learn which objects (the blickets) have a novel hidden causal power to activate a machine (the blicket detector). Typically, even 4-year-olds require only a handful of observations in order to learn about the existence of this novel causal relation. Moreover, they use this knowledge both to make predictions and to design novel interventions and counterfactuals in much the same way that the causal graphical models formalism would suggest (Gopnik et al. 2004). We use this setting to test the hypothesis that adults and children integrate prior knowledge and covariational evidence about causal relations in a way that is consistent with Bayesian inference. We explore two different kinds of prior knowledge. First, we look at the assumptions about the probability that an object is a blicket. Second, we explore a more abstract assumption about the functional form of the causal relations participants observe: whether they are deterministic or probabilistic. Our model allows us to integrate both these forms of prior knowledge with current evidence. We then examine the consequences of Bayesian inference and causal learning 5 modifying these assumptions through experiments in which we change the probability with which causal relations exist and whether those relations are deterministic or probabilistic. Our strategy of conducting experiments with both adults and children illustrates the generality of our formal approach, and provides the opportunity to investigate causal induction where it is easiest to study and where it is most important. Adult participants are willing to answer a variety of questions about causality and produce multiple numerical ratings, resulting in data that are sufficiently fine-grained to allow quantitative evaluation of our models. While we can obtain only a relatively coarse characterization of the beliefs of children, they are arguably the group whose behavior we would most like to understand. Four-year-olds are still in the process of forming their deepest theories of the causal structure of their world, and using their capacity for causal induction to do so. Conducting parallel experiments with both groups provides the opportunity to test the details of our models, and to show how they might help us understand the mechanisms of cognitive development, particularly because causal graphical models explain many facets of children’s causal reasoning. Further, there is a long literature on causal reasoning in young children, suggesting basic perceptual and reasoning abilities are in place at quite young ages (e.g., Bullock et al., 1982; Carey, 2009; Leslie & Keeble, 1987; Shultz, 1982), and comparing children to adults makes it possible to identify aspects of causal reasoning that might develop over time. The Bayesian approach to causal induction that we test follows in a long tradition of formal models of human judgments about causal relations (e.g., Ward & Jenkins, 1965; Shanks, 1995; Cheng, 1997. Previous models focus on covariation between cause and effect as the basis for evaluating causal relations, and are usually applied to experiments in which such covariation is expressed over many trials on which causes and effects might occur. Our Bayesian inference and causal learning 6 experiments present a challenge to these models, showing that adults and children can learn causal relations from few observations, and that situations in which people observe exactly the same covariational evidence lead to different conclusions when people have different prior knowledge about the causal relations involved. The plan of the paper is as follows. We first review formal approaches to human causal induction, and introduce the key ideas behind our Bayesian approach. We then discuss how this approach can incorporate prior knowledge on the part of the learner, and how appropriate knowledge can make it possible to learn causal relations from small amounts of data. Showing that this approach can account for some of the basic results using the blicket detector paradigm motivates our experiments. Experiments 1-3 explore the consequences of manipulating the probability that a causal relation exists in this paradigm. Experiments 4-5 examine how adults and children integrate more abstract prior knowledge with the evidence they observe by considering inferences when the mechanism between causes and effects is deterministic or probabilistic. We then consider some of the implications of these results and the limitations of our analysis in the General Discussion. Formal models of causal induction Most formal models of causal induction follow one of two approaches: taking causal induction to be a form of associative learning, or defining a rational procedure for estimating the strength of a causal relation. We will briefly review these two approaches. Learning associations One way that an individual could reason about causal relations among events in the world is to note their co-occurrence. Several psychologists have proposed that causal induction is based on recognizing associations among events (e.g., Cramer et al., 2002; Bayesian inference and causal learning 7 Dickinson & Shanks, 1995; Pearce, 1987). Many of these models have their origin with the Rescorla-Wagner (1972) model, which increases the strength of association between a cue and an outcome when the cue is present on a trial where the outcome occurs unexpectedly. This model provides an account of many basic findings in the associative learning literature, including Kamin’s (1968) “blocking” phenomenon, in which a novel cue does not become associated with an outcome when that cue only appears in the presence of another cue that has already been associated with the outcome. Subsequent research in associative learning has uncovered phenomena that cannot be captured easily by simple associative models. One of these phenomena is backwards blocking, a procedure that is identical in its contingencies to blocking, but reverses the order in which the trials are presented (Chapman, 1991; Kruschke & Blair, 2000; Miller & Matute, 1996; Shanks, 1985). In the first part of training, two cues (A and B) occ