Inferential Dependencies in Causal Inference: A Comparison of Belief-Distribution and Associative Approaches Christopher D. Carroll, Patricia W. Cheng, and Hongjing Lu University of California, Los Angeles Causal evidence is often ambiguous, and ambiguous evidence often gives rise to inferential dependen- cies, where learning whether one cue causes an effect leads the reasoner to make inferences about whether other cues cause the effect. There are 2 main approaches to explaining inferential dependencies. One approach, adopted by Bayesian and propositional models, distributes belief across multiple expla- nations, thereby representing ambiguity explicitly. The other approach, adopted by many associative models, posits within-compound associations—associations that form between potential causes—that, together with associations between causes and effects, support inferences about related cues. Although these fundamentally different approaches explain many of the same results in the causal literature, they can be distinguished, theoretically and experimentally. We present an analysis of the differences between these approaches and, through a series of experiments, demonstrate that models that distribute belief across multiple explanations provide a better characterization of human causal reasoning than models that adopt the associative approach. Keywords: causal inference, inferential dependencies, retrospective revaluation, cue competition Causal evidence is often ambiguous. When trying to identify the cause of a recent illness, the reason why a friend failed to return a phone call, or the cause of a car accident, possible explanations abound. In such situations, subsequent learning about one of the possible causes may support inferences about the other possible causes. Consider, for example, a traveler who becomes ill after a flight where he ate a suspect meal and sat next to a coughing passenger. His illness may have been caused by the meal or by his coughing neighbor. After learning that other passengers who ate the inflight meal did not become ill, the traveler would probably conclude that the cause was his coughing neighbor. In such cir- cumstances, it seems as if the traveler retrospectively reevaluates the ambiguous initial evidence (the two plausible causes of his illness) in light of the subsequent evidence (the harmlessness of the inflight meal). Consequently, the inference is said to involve retrospective revaluation (e.g., Van Hamme & Wasserman, 1994). More generally, we say that there is an inferential dependency between two possible causes when learning about one of them can support an inference regarding the other. How should we explain inferential dependencies in causal rea- soning? There are two main approaches to the problem. The associative approach explains inferential dependencies by utilizing within-compound associations—associations that form between potential causes, in addition to the typical associations between each cause and its effect (e.g., Dickinson & Burke, 1996; Stout & Miller, 2007; Van Hamme & Wasserman, 1994). Within- compound associations are assumed to form when potential causes co-occur, as is typically the case when there is confounding and thus the evidence is ambiguous. The association between co- occurring cues—say, potential causes c 1 and c 2 —allows the weight of the association between c 1 and the effect e to be updated for events (trials) on which c 1 is absent; when c 2 occurs, its activation can activate c 1 via the within-compound association. This is unlike in typical associative models, in which only cues that are present are activated and eligible for updating. The within- compound association thereby provides a representation for ex- plaining inferential dependencies in situations involving ambigu- ity. For example, an associative model might posit that the traveler’s meal and the coughing neighbor are associated through a within-compound association. The within-compound association could be used to support the inference that, if one of the cues is not causal, then the other should be. Note, however, that at any given moment, an associative network, regardless of whether it supports within-compound associations, is in a single state where each associative strength is estimated by a single value. Thus, various alternative explanations of ambiguous evidence would have to map onto the same state of an associative network. In other words, the approach does not provide a means for repre- senting multiple explanations at the same time. For example, consider a series of trials in which two cues, A and B, simultane- ously occur and a target effect follows. In an associative network, it seems reasonable to represent this ambiguous evidence by a state where each of the two cues has a cue– effect association with This article was published Online First September 10, 2012. Christopher D. Carroll and Patricia W. Cheng, Department of Psychol- ogy, University of California, Los Angeles; Hongjing Lu, Departments of Psychology and Statistics, University of California, Los Angeles. The preparation of this article was supported by Air Force Office of Scientific Research Grant FA 9550-08-1-0489 to Alan Yuille and Patricia W. Cheng. Preliminary reports of this research were presented at the 32nd and 33rd Annual Conferences of the Cognitive Science Society. We thank David Danks for extremely helpful comments, and we thank Betty Huang and Aaron Placensia for assistance with data collection. Correspondence concerning this article should be addressed to Christo- pher D. Carroll, who is now at Department of Psychology, Carnegie Mellon University, Pittsburgh, PA 15213. E-mail: cdcarroll@gmail.com This document is copyrighted by the American Psychological Association or one of its allied publishers. This article is intended solely for the personal use of the individual user and is not to be disseminated broadly. Journal of Experimental Psychology: General © 2012 American Psychological Association 2013, Vol. 142, No. 3, 845– 863 0096-3445/13/$12.00 DOI: 10.1037/a0029727 845