Preschoolers sample from probability distributions

Preschoolers sample from probability distributions Stephanie Denison (smdeniso@berkeley.edu) Elizabeth Baraff Bonawitz (liz b@berkeley.edu) Alison Gopnik (gopnik@berkeley.edu) Thomas L. Griffiths (tom griffiths@berkeley.edu) Department of Psychology, University of California, Berkeley, Berkeley, CA 94720 USA Abstract Researchers in both educational and developmental psychol- ogy have suggested that children are not particularly adept hy- pothesis testers, and that their behavior can often appear irra- tional. However, a growing body of research also suggests that people do engage in rational inference on a variety of tasks. Recently researchers have begun testing the idea that reasoners may be sampling hypotheses from an internal probability dis- tribution when making inferences. If children are reasoning in this way, this might help to explain some seemingly irrational behavior seen in previous experiments. Forty 4-year-olds were tested on a probabilistic inference task that required them to make repeated guesses about which of two types of blocks had been randomly sampled from a population. Results suggest that children can sample from a probability distribution as evi- denced by the fact that, as a group, they engaged in probability matching and that the dependency between successive guesses decreased over time. Keywords: Cognitive Development; Causal Learning; Ap- proximate Bayesian Inference; Sampling Hypotheses Introduction Young children are faced with a variety of novel situations on a daily basis. They encounter countless episodes in which they must reason about why particular events unfold the way they do, what this means in terms of how related events might unfold in the future, and how this newly acquired informa- tion fits into the knowledge they already possess. Humans revise their beliefs throughout development, often beginning with relatively flawed beliefs and progressing towards an in- creasingly accurate portrayal of the world. However, no cur- rent theory provides a satisfactory explanation of how chil- dren decide which hypotheses to test. Somehow they must search through the potentially infinite number of hypotheses that exist at the beginning of the learning process. Here, we investigate this question by asking whether young children can make probabilistic inferences via a process of sampling hypotheses from probability distributions. The question of whether children and adults are capable of using rational inference to search through a hypothesis space and revise their beliefs has drawn mixed empirical findings. To begin, Piaget noted that children tend not to reason sys- tematically about hypotheses, at least until they reach the for- mal operational stage in late childhood (Piaget, 1983). Since Piaget, some researchers have found evidence to corroborate this claim, stating that children often appear to navigate ran- domly through a selection of predictions and explanations (Siegler & Chen, 1998). For example, researchers in edu- cational psychology have revealed evidence suggesting that young children and even non-expert adults are not particu- larly skilled hypothesis testers (e.g., Kuhn, 1989; Klahr, Fay, & Dunbar, 1993). Furthermore, developmental psychologists have found that children often revise their beliefs surprisingly slowly, suggesting a struggle to efficiently update theories (e.g., Carey, 1991; Wellman, 1990). On the other hand, at least two kinds of evidence exist to suggest that children might be capable of using rational inference to generate, search through and evaluate hypothe- ses. First, recent research in cognitive psychology suggests that people reason in ways that are consistent with optimal Bayesian models in a variety of tasks (e.g., Griffiths & Tenen- baum, 2005; Goodman, Tenenbaum, Feldman, & Griffiths, 2008). Although most of this work examines adult reason- ing, a growing body of evidence suggests that children can also reason in a way that is consistent with Bayesian infer- ence (e.g., Gopnik et al., 2004; Kushnir & Gopnik, 2005; Schulz & Gopnik, 2004; Schulz, Bonawitz, & Griffiths, 2007; Goodman et al., 2006). For example, Xu and Tenenbaum (2007) found that preschoolers can systematically integrate prior knowledge regarding hierarchical information with ev- idence in order to apply the correct labels to a variety of ob- jects in a word learning task and Schulz et al. (2007) and Kushnir and Gopnik (2007) found that children’s causal in- ferences rationally depend on both their prior beliefs and the observed evidence. Second, many researchers advocate the theory-theory of conceptual development, which states that children’s knowledge is organized into abstract, coherent con- ceptual systems, similar to those found in science (Carey, 1985; Gopnik & Meltzoff, 1997; Murphy & Medin, 1985; Wellman & Gelman, 1992). This framework predicts that children will engage in hypothesis testing in ways similar to scientists during learning, and much evidence has accu- mulated in support of this view (e.g. see Karmiloff-Smith & Inhelder, 1974; Bonawitz, Lim, & Schulz, 2007; Legare, Gelman, & Wellman, in press). However, the theory-theory does not specify where the hypotheses are derived from in the first place or how children could be expected (albeit un- consciously) to compute full Bayesian inference over (often) infinite hypothesis spaces. The sampling hypothesis Although Bayesian inference corresponds well to the theory- theory of conceptual development, researchers who advocate a rational approach to human inference do not suggest that adults and children actually work through the steps of Bayes’ rule in daily life. Evaluating all possible hypotheses each time new data are observed would not be feasible both from a for- mal and a practical standpoint, given the large number of hy- potheses that would require consideration. One way to think about how the mind may be approximating Bayesian infer- ence is to start with good engineering answers to this prob- lem. Techniques for approximating Bayesian inference have