Simplicity and Goodness-of-Fit in Explanation: The Case of Intuitive Curve-Fitting

Simplicity and Goodness-of-Fit in Explanation: The Case of Intuitive Curve-Fitting Samuel G. B. Johnson (samuel.johnson@yale.edu) Department of Psychology, Yale University 2 Hillhouse Ave., New Haven, CT 06520 USA Andy Jin (ajin@live.unc.edu) Department of Psychology, University of North Carolina, Chapel Hill 235 E. Cameron Ave., Chapel Hill, NC 27599 USA Frank C. Keil (frank.keil@yale.edu) Department of Psychology, Yale University 2 Hillhouse Ave., New Haven, CT 06520 USA Abstract reasoning (Lombrozo, 2007). The posterior odds favoring the simple explanation [H 1 ] over the complex explanation [H 2 ,H 3 ] is the product of the prior odds [P(H 1 )/P(H 2 ,H 3 )], representing the probability of each hypothesis being true before observing the evidence, and the likelihood ratio [P(E 1 ,E 2 |H 1 )/P(E 1 ,E 2 |H 2 ,H 3 )], representing the probability of the evidence being observed under each hypothesis. If we assume that the simple explanation (influenza) and the complex explanation (hay fever and Strep throat) lead to both symptoms with the same probability, then the likelihood ratio is 1. Lombrozo (2007) noted, however, that the prior odds often favor simple explanations: If H 1 , H 2 , and H 3 have equal base rates, then P(H 2 ,H 3 ) = [P(H 1 )] 2 . Since the complex explanation has a lower prior, the posterior odds favor the simpler explanation. However, the posterior odds do not always favor the simpler explanation, because the evidence is often made more likely (or, put differently, is “better modeled”) by a complex explanation than by a simple explanation, leading the likelihood ratio to favor the complex explanation. For example, consider a patient who has a runny nose and a stomach ache. One explanation is that the patient has only a cold, which would make a runny nose very likely and would make a stomach ache plausible (but not especially likely). Alternatively, the patient could have both a cold and a stomach virus, which would make both symptoms very likely. The more complex explanation has a lower prior, but a higher likelihood. This is often the case: The more (generative) causes invoked by an explanation, the lower its prior, but the more likely it is to predict the observed effects because each added cause has the potential to produce the observed effects. How do people negotiate this trade-off between simplicity and goodness-of-fit? One possibility is that people would require disproportionate evidence to abandon a simpler explanation in favor of a more complex one. Lombrozo (2007) found that people use a simplicity heuristic, Other things being equal, people prefer simpler explanations to more complex ones. However, complex explanations often provide better fits to the observed data, and goodness-of-fit must therefore be traded off against simplicity to arrive at the most likely explanation. In three experiments, we examine how people negotiate this trade- off. As a case study, we investigate laypeople’s intuitions about curve-fitting in visually presented graphs, a domain with established quantitative criteria for trading off simplicity and goodness-of-fit. We examine whether people are well-calibrated to normative criteria, or whether they instead have an underfitting or overfitting bias (Experiment 1), we test people’s intuitions in cases where simplicity and goodness-of-fit are no longer inversely correlated (Experiment 2), and we directly measure judgments concerning the complexity and goodness-of-fit in a set of curves (Experiment 3). To explain these findings, we posit a new heuristic: That the complexity of an explanation is used to estimate its goodness-of-fit to the data. Keywords: Explanation; causal reasoning; statistics; simplicity; diagnostic reasoning. intuitive Introduction The simplest explanation is often the most likely to be true. Suppose, for example, that a patient has two symptoms: a runny nose (E 1 ) and a cough (E 2 ). Three diagnoses are available: influenza (H 1 ), which explains both symptoms; Hay fever (H 2 ), which explains only the runny nose; and Strep throat (H 3 ), which explains only the cough. If these diseases are equally common, then any reasonable doctor would surely posit influenza (H 1 ) to explain both symptoms, rather than hay fever to explain the runny nose and Strep throat to explain the cough (H 2 & H 3 ). Laypeople share these intuitions: Other things being equal, adults and children prefer explanations invoking fewer causes and believe they are more likely to be true (Bonawitz & Lombrozo, 2012; Lombrozo, 2007). These intuitions relate to normative probabilistic