Simplicity Bias in the Estimation of Causal Functions

Simplicity Bias in the Estimation of Causal Functions Daniel R. Little (drlittle@indiana.edu) Richard M. Shiffrin (shiffrin@indiana.edu) Department of Psychological and Brain Sciences, Indiana University 1101 E. 10th St, Bloominton IN 47405 USA Abstract Kalish, Lewandowsky, & Kruschke, 2004). Typically, the focus in function learning is on the type and relative diffi- culty of learning different functions; most function learning studies have not directly examined prior beliefs. However, there have been some recent investigations of function biases in the iterated learning paradigm (see e.g., Griffiths, Kalish, & Lewandowsky, 2008; Kalish, Griffiths, & Lewandowsky, 2007). In an iterated learning task, participants are trained on stimuli drawn from the learning outcomes of the previous participant. Each learner is assumed to have a set of hypothe- ses about the causal relationship between the input and output variables. A rational learner assigns probabilities to each hy- pothesis in accordance with the posterior probability of that hypothesis given the data. According to Bayes’ rule the pos- terior probability, p(h|d), is: We ask observers to make judgments of the best causal func- tions underlying noisy test data. This method allows us to ex- amine how people combine existing biases about causal rela- tions with new information (the noisy data). Participants are shown n data points representing a sample of noisy data from a supposed experiment. They generate points on what they believe to be the true causal function. The presented functions vary in noise, gaps, and functional form. The method is similar to function learning studies, but minimizes the roles of learn- ing and memory. To what degree do the participants exhibit a bias for simple linear functions? We describe a hierarchical Bayesian polynomial regression model to quantify complex- ity. The results show the expected bias for simplicity, but with some interesting individual differences. Keywords: causal models, function learning; prior knowl- edge; hierarchical Bayesian regression. In science generally, in statistical inference, in studies of model selection, and in psychological theorizing, we design our inference approaches to trade off fit to observed data (models are good that fit well) and complexity (models or explanations that fit or explain everything are bad). In the present research we explore how observers form causal mod- els for noisy data by asking observers to estimate functions as explanations for a set of noisy data points. We ask: How do mental causal models balance fit and complexity? Fur- ther, how does the balance of fit and complexity achieved by observers compare with the balance imposed by a rational system? We explore the former question by fitting Bayesian hierarchical models to the causal functions produced by the observers. The latter question is addressed by comparing the model for the observers to a Bayesian hierarchical model fit to the presented noisy data. Recent research into beliefs about casual relationships has demonstrated that people prefer simple explanations over more complex explanations (see e.g., Lombrozo, 2006, 2007). In fact, simplicity biases have been forwarded as underlying many fundamental cognitive processes (Chater & Vitanyi, 2003). Here we are primarily concerned with how people form mental models summarizing relationships between continuous input and output quantities (i.e., data). Learning about continuous variables has typically been stud- ied as a function learning phenomena. Within this paradigm, input values from a single function are displayed one at a time, the participant responds with an output value and re- ceives corrective feedback about the true output value. Over time, participants form some representation about the un- derlying functional relationship between the two variables (i.e., exemplar-based associations, DeLosh, Busemeyer, & McDaniel, 1997; or a mixture of linear function ’experts’, p(h|d) = p(d|h)p(h) p(d) where p(d|h) is the probability or likelihood of the data given a hypothesis, p(h) is the prior probability of that hy- pothesis and p(d) is the marginal probability of the data, ∑ i p(d|h i )p(h i ). In an iterated function learning task, re- sponses tend to converge to a positive linear function indicat- ing a bias towards positive linear functions in people’s prior beliefs about functions(Kalish et al., 2007). Non-Bayesian approaches to function learning also pro- vide hints of a simplicity bias. Humans learn positive lin- ear functions faster than negative linear functions or non- linear functions (Carroll, 1963). One successful theoretical approach assumes that performance is driven by combining a pre-existing set of linear functions (i.e., POLE; Kalish et al., 2004). However, the representations formed in function learning studies are determined in part by cognitive limita- tions on, and the processes of, attention, learning, and mem- ory. For example, the relation between an input value and output value presented at one point in time might be partially or wholly forgotten later, and possibly distorted through in- ference in the direction of simplicity. As another example, extreme points might be attended and remembered best, lead- ing to a different set of distortions. In the current paper, we present an experimental method for studying function repre- sentations in which all the data is presented simultaneously, in a form easy to perceive and process. In this method, one should not have to parcel out the effects due to faulty mem- ory for the data points on the function, or inadequate learn- ing of the function over time. We use the method to address two related questions: do people show a bias towards sim- ple (possibly linear) functional relationships when faced with