Uncertainty and dependency in causal inference

Uncertainty and dependency in causal inference Christopher D. Carroll (cdcarroll@ucla.edu) Department of Psychology, UCLA Patricia W. Cheng (cheng@lifesci.ucla.edu) Department of Psychology, UCLA Hongjing Lu (hongjing@ucla.edu) Department of Psychology, UCLA Abstract When inferring causal relationships, people are often faced with ambiguous evidence. Models of causal inference have taken different approaches to explain reasoning about such evidence. One approach – epitomized by Bayesian models of causal inference – defers judgment by representing uncertainty across multiple explanations. Another approach – usually adopted by associative models – approximates uncertainty by positing within-compound associations, a special type of association that forms between simultaneously presented cues. Although these approaches explain many of the same experimental findings, we note some limitations of the latter approach. Within-compound associations form whenever two cues are presented simultaneously – even when the causal influences of the cues are already known. Since associative models use within-compound associations to modify beliefs about one potential cause when learning about another, associative models therefore predict that cues with known causal influences can have their influence revised as a result of being presented with other cues. In two experiments, we tested the predictions of the two approaches. The results were consistent with models that represent uncertainty across multiple explanations and inconsistent with models that use within-compound associations. Keywords: causal reasoning; causal inference; uncertainty; associative models; Bayesian models Introduction Everyday causal inference often requires reasoning about ambiguous evidence. Consider a reasoner who is trying to explain events such as a recent illness, a lapse in a friendship, or a car accident. Each of these events has many possible explanations, and, in many cases, the reasoner will not be able to identify the correct explanation with certainty. While there are different ways to respond to ambiguous evidence, the most reasonable response involves deferring judgment by representing the uncertainty associated with the evidence. That is, rather than committing to a single explanation prematurely, a reasoner presented with ambiguous evidence should distribute his or her belief across the possible explanations in accordance with the plausibility of each explanation. It has commonly been observed that while Bayesian models exemplify this sort of approach, associative models adopt another approach (e.g., Courville, Daw, Gordon, & Touretzky, 2003; Kruschke, 2008; Lu, Rojas, Beckers, & Yuille, 2008; Sobel, Tenenbaum, & Gopnik, 2004). Most of the research that investigates reasoning about ambiguous evidence focuses on situations where there is an inferential dependency such that learning about whether one cue causes the effect provides information about whether other cues cause the effect. Consider, for example, a situation where the effect occurs in the presence of two possible causes (we write this as AB+, letting letters represent the potential causes and +/– represent the presence/absence of the effect). This ambiguous evidence establishes an inferential dependency between cues A and B because subsequent learning about cue A can provide information about cue B and vice versa (e.g., if shown A– trials, a reasoner would probably conclude that cue B definitely causes the effect). As we will see later, Bayesian models can explain this inferential dependency – as well as other inferential dependencies – by distributing belief across multiple explanations. The evidence for the superiority of this approach, however, is less strong than one might expect: some associative models – which do not seem to defer judgment or represent uncertainty – explain the same experimental findings. At the very least, associative models are able to approximate a genuine representation of uncertainty and handle some types of ambiguous evidence. In the present paper, we present (1) an analysis of the weakness of the associative approximation of uncertainty and (2) an empirical test based on the analysis that clearly differentiates the two approaches. Associative models The Rescorla-Wagner (RW) model (Rescorla & Wagner, 1972) is the most well-known associative model. The RW model adopts the following learning rule, which modifies the associations between a cue (potential cause) and the effect in order to reduce prediction error: In this learning rule, s i represents the salience or learning rate for cue i when it is present (s i = α) or absent (s i = 0), s e represents the salience of the effect (s e = β), T represents the presence (T = 1) or absence (T = 0) of the effect, and V i represents the prior association between cue i and the effect. The summation, which occurs over all cues present on a given trial, represents the predicted strength of the effect. The difference between T and the summation therefore