Hierarchical Bayesian models as formal models of causal reasoning

knowledge can come in various forms, and it may form complex hierarchies. The most abstract form of knowledge may be called causal principles, which probably include the assumption that causes precede their effects, that nothing happens without a cause, and that causes generate their effect unless prevented by an inhibitory factor (Audi 1995). A fundamental assumption might also be that a manipulation of a cause changes its effects but not vice versa (Woodward 2003). People seem to automatically consider these assumptions in causal learning (Waldmann 1996). For example, research on the induction of hidden causes has shown that people assume the presence of a hidden cause when they observe an effect that cannot be explained otherwise (Hagmayer and Waldmann 2007; Luhmann and Ahn 2007). Even young children infer that objects having the same observable effect will have the same internal, hidden property causing the effect (Sobel, Yoachim, Gopnik, Meltzoff, and Blumenthal 2007). Structural principles like trees, causal chains or loops are also highly abstract forms of theoretical knowledge. Research on property induction, for example, has shown that people consider a taxonomic tree structure when judging whether a genetic property of one species can be found in another species. By contrast, people consider causal chains when they make inferences with respect to properties that can be transferred through food chains (e.g. infections, toxins) (Tenenbaum, Kemp, Griffiths, and Goodman 2011). When predicting the effects of interventions in complex systems, people often assume causal chains although causal feedback loops would be more appropriate (Sterman 2000). For example, people expect antibiotics to eliminate bacterial infections, despite the fact that antibiotics increase the bacteria’s resistance which may offset the desired effect in the long run. 40 Y. Hagmayer and R. Mayrhofer Causal schemata are assumptions about how multiple causes may interact (Kelley 1973). Two prominent schemata are multiple sufficient causes entailing that various causes may generate an effect on their own and multiple necessary causes, which entail that a certain set of causes has to be present for the effect to occur. However, people may also learn that an effect is only generated when an individual cause is present (see, Lucas and Griffiths 2010). Furthermore, people generally seem to expect that effects of separate causes add up (Beckers, De Houwer, Pineño, and Miller 2005) and that causes act independently (Mayrhofer, Nagel, and Waldmann 2010; see also Novick and Cheng 2004). Waldmann (2007) showed that for extensional properties (e.g. different causal strengths) people assume effects to add up, while for intensional properties (e.g. degrees of liking) people expect effects to average. A causal framework theory is the assumption that observable causal relations among particular events or objects (tokens) are instantiations of more general causal laws connecting types of events or objects (Kemp, Goodman, and Tenenbaum 2010). Such a framework theory entails that events or objects of the same type (i) generate the same effects with the same likelihood, and (ii) share the same features. (We will describe a respective HBM in the next section of the paper.) Note that none of these highly abstract theories includes any assumptions about the entities being causally related or the nature of the causal relation. Theoretical knowledge, however, could also be domain-specific. It may include assumptions about the causally relevant entities, their properties and potential relations (Griffiths and Tenenbaum 2009). Research in developmental psychology has shown that infants and young children already have domain-specific expectations about causal entities and their relations (Carey 2009). For example, while children expect temporal and spatial contiguity to be necessary for causation in a mechanical system, they accept causation at a distance when the agents are persons. Finally, theoretical assumptions may concern the nature of the causal mechanisms. Many studies have shown that people draw different inferences from the same observations depending on whether they know about a potential mechanism underlying the observed relation (Ahn, Kalish, Medin, and Gelman 2000). For example, a moderate statistical relation was seen as good evidence for causation when a plausible mechanism was known, while this was not the case when no mechanism could be envisioned (Koslowski 1996). Developmental research found that children being familiar with causation via electric wiring did not require spatio-temporal contiguity to infer causality while younger children did (Bullock, Gelman, and Baillargeon 1982). Research also shows that people have a hard time to induce causal models, when they have very little abstract knowledge (e.g. Steyvers, Tenenbaum, Wagenmakers, and Blum 2003; see also Lagnado, Waldmann, Hagmayer, and Sloman 2007). In general, performance in causal induction increases substantially, when people can observe temporal relations or can intervene (Lagnado and Sloman 2004). Research on the control of complex systems shows that participants often gain very limited knowledge about the underlying causal structure despite extensive learning experience (Osman 2010). HBMs can explain this finding as the available evidence in these studies was often compatible with numerous causal hypotheses, and participants normally lacked domain specific knowledge (see, Hagmayer, Meder, Osman, Mangold, and Lagnado 2010, for a more detailed discussion). In sum, there are many forms of abstract theoretical causal knowledge that may guide causal induction. This abstract knowledge can be domain-general or domain-specific. A growing body of evidence shows that such abstract causal knowledge affects causal induction. 4. Induction of causal laws Causal relations may hold between particular objects or event tokens, but also between types of causes and effects. Causal relations on a type level have been called causal laws (Schulz, Argument and Computation 41 Figure 2. Hierarchical Bayesian model of category learning and causal induction. Goodman, Tenenbaum, and Jenkins 2008). Causal laws specify the structure and strength of the causal relations between types of causes and types of effects. To induce causal laws, both categories of causes and effects as well as the causal relations between them have to be learnt. HBMs allow us to model these inferences (Kemp et al. 2010). Figure 2 provides an illustration. On the most abstract level, there is a framework theory assuming that there are causal relations between types of causes and effects without specifying what these types are. There is also the assumption that tokens of the same type share certain features. On the level of causal laws are causal relations between cause and effect types. Representations of causal relations between types and their features can also be found on this level. Causal models are characterised by causal relations between a type and a token. Data are observations of single causal relations between token causes and token effects and observations of the features of cause and effect tokens. Some research has been conducted investigating the induction of categories during causal learning. Lien and Cheng (2000) presented participants with objects having different colours and sizes which caused an effect with a certain probability. By manipulating the probabilities, they were able to show that participants induced categories of objects (e.g. warm-coloured versus cold coloured objects) that were optimal for predicting the effect. Crucially, participants used the new categories to predict the effect of objects they had not seen before. Kemp et al. (2010) constructed a HBM which was able to reproduce the findings of a simplified version of the studies by Lien and Cheng (2000). Waldmann, Hagmayer and colleagues (Waldmann and Hagmayer 2006; Waldmann, Meder, von Sydow, and Hagmayer 2010) extended this research and showed that both causes and effects are spontaneously categorised according to the causal relations they are part of, and that these categories are transferred to later learning episodes even when they do not allow for optimal predictions. This transfer, however, seems to depend on people’s abstract causal knowledge (Hagmayer, Meder, von Sydow, and Waldmann 2011). Categories of causes are only used to predict novel effects if these new effects are plausible effects of the type of cause (i.e. if there is a plausible causal mechanism). Abstract structural knowledge also seems to be important. Categories based 42 Y. Hagmayer and R. Mayrhofer on one causal relation were transferred to a second causal relation only if the second relation was connected to the first relation forming a continuous causal chain (Hagmayer et al. 2011). Otherwise people preferred to induce separate categories for each causal relation. Developmental research shows that even pre-schoolers use the causal effects to categorise objects (Gopnik and Sobel 2000) and that these categories may even override perceptual differences between objects (Nazzi and Gopnik 2003). Later studies indicate that pre-schoolers even have the ability to abstract causal laws (Schulz et al. 2008). Children in these studies categorised coloured blocks with respect to their capacity to generate certain sounds when touching other types of blocks. Crucially, children used the observation of one effect to categorise an object and then use the category to make predictions about another effect. HBMs can account for this transfer effect (Schulz et al. 2008).