Causal Models Interact with Structure Mapping to Guide Analogical Inference

Causal Models Interact with Structure Mapping to Guide Analogical Inference Hee Seung Lee (heeseung@ucla.edu) Keith J. Holyoak (holyoak@lifesci.ucla.edu) Department of Psychology, University of California, Los Angeles Los Angeles, CA 90095 USA Abstract We recently proposed a theoretical integration of analogical transfer with causal learning and inference (Lee & Holyoak, 2008). A Bayesian theory of learning and inference based on causal models (Lee, Holyoak & Lu, 2009) accounts for the fact that judgments of confidence in analogical inferences are partially dissociable from measures of the quality of the mapping between source and target analogs. The integrated theory postulates a dual role for causal relations, which can guide both analogical mapping and also subsequent inferences about the target. It follows that depending on whether or not a mapping is structurally ambiguous, dropping a preventive cause from the target can either decrease or increase confidence in the same analogical inference. We report an experiment that yielded data in close agreement with predictions of the Bayesian theory. These results provide further support for the importance of integrating analogical transfer with the broader framework of causal models. Keywords: analogical inference; causal models; mapping; Bayesian modeling. Introduction Analogical transfer plays a key role in scientific reasoning (Dunbar & Fugelsang, 2005). Indeed, in some areas of science in which experimental research is impossible, such as historical ethnography, analogy may provide the only viable mechanism for evaluating hypotheses. Talalay (1987) gives the example of interpreting the function of strange clay fragments discovered in Neolithic Greek sites: individual female legs, apparently never attached to torsos, that had been manufactured in pairs and later broken apart. The best clues to their function have come from other cultures in which similar tokens are known to have served to seal contracts and provide special evidence of the identity of the bearer (in feudal China, for example, a valuable piece of jade would be broken in two to mark a contract between a master and his vassal, with each keeping one piece so they could later be matched). Here the known function in a source domain has a causal connection to the form of relevant artifacts, and the ethnographer makes the analogical inference that a similar cause may have operated in the target domain (see Bartha, 2010). The general question faced by a reasoner using analogy to make inferences is: Given prior knowledge at various levels of abstraction, including one or more examples that serve as source analogs, what is the probability that any potential inference about a target analog is true? For analogical inferences that involve empirical claims about the world (e.g., scientific hypotheses), answering this question depends on at least two basic subprocesses: deciding how the causally-relevant elements of the source analog(s) relate to elements of the target (structure mapping), and using the corresponding causal relations suggested for the target to estimate the probabilities of potential inferences about the target (causal inference). In the above ethnography example, mapping is required to relate the two pieces of a broken jade object to the two parts of a broken piece of pottery; causal inference is required to evaluate the probability that the clay fragments could achieve a function analogous to that achieved by a divided jade object. Both structure mapping and causal inference have received considerable attention within cognitive science. Mapping has been the central focus of many models of analogical reasoning (e.g., Falkenhainer, Forbus & Gentner, 1989; Holyoak & Thagard, 1989; Hummel & Holyoak, 1997). Causal learning and inference have also been studied extensively, with theoretical work largely based on the framework of causal models (Pearl, 1988; Waldmann & Holyoak, 1992; Waldmann, Holyoak & Fratianne, 1995). The power PC theory (Cheng, 1997) provides a quantitative explanation of how the strengths of probabilistic causal relations can be learned from contingency data. More recently, this theory has been extended based on a Bayesian formulation (Griffiths & Tenenbaum, 2005; Lu et al., 2008). Theories of category-based induction have also been enriched by adopting the framework of causal models (e.g., Ahn, 1999; Kemp, Goodman & Tenenbaum, 2007; Sloman, 1994; Rehder, 2009). Integrating analogical inference with causal models We have proposed that a more complete understanding of analogical transfer requires specifying the role played by causal models (Lee & Holyoak, 2008; Lee, Holyoak & Lu, 2009). Figure 1 schematizes causal models for a source (left) and target analog (right). The nodes represent variable causes (C) and effects (E). The superscripts ( S , T ) indicate the source and the target, respectively. The links represent the causal structure (only linked nodes have direct causal connections). The vectors w i represent the causal polarity (generative or preventive) and the causal strength for links. A key assumption is that analogical transfer involves using causal knowledge of the source to develop a causal model of the target, which can in turn be used to derive a variety of inferences about the values of variables in the target. Causal relations in Bayesian causal models can carry information about existence of causal links (e.g., causal structure) and distributions of causal strength, as well as about the generating function by which multiple causes combine to influence effects.