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), whereas mechanism-based approaches focus on the role of prior knowledge about causal mechanisms (e.g., Ahn & Kalish, 2000; Bullock, Gelman, & Baillargeon, 1982; Shultz, 1982; Wolff, 2007). In this article, 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 domain-general statistical inference guided by domain-specific prior knowledge. We show how covariational evidence and prior knowledge about causal mechanisms can be combined via 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 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.

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, Tenenbaum, & Gopnik, 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 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 it 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 substantial 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., Cheng, 1997; Shanks, 1995; Ward & Jenkins, 1965). Previous models focus on covariation between cause and effect as the basis for evaluating causal relations, and they are usually applied to experiments in which such covariation is expressed over many trials on which causes and effects might occur. Our 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 article 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 and 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.