Fast and Accurate Learning When Making Discrete Numerical Estimates

Abstract

Many everyday estimation tasks have an inherently discrete nature, whether the task is counting objects (e.g., a number of paint buckets) or estimating discretized continuous variables (e.g., the number of paint buckets needed to paint a room). While Bayesian inference is often used for modeling estimates made along continuous scales, discrete numerical estimates have not received as much attention, despite their common everyday occurrence. Using two tasks, a numerosity task and an area estimation task, we invoke Bayesian decision theory to characterize how people learn discrete numerical distributions and make numerical estimates. Across three experiments with novel stimulus distributions we found that participants fell between two common decision functions for converting their uncertain representation into a response: drawing a sample from their posterior distribution and taking the maximum of their posterior distribution. While this was consistent with the decision function found in previous work using continuous estimation tasks, surprisingly the prior distributions learned by participants in our experiments were much more adaptive: When making continuous estimates, participants have required thousands of trials to learn bimodal priors, but in our tasks participants learned discrete bimodal and even discrete quadrimodal priors within a few hundred trials. This makes discrete numerical estimation tasks good testbeds for investigating how people learn and make estimates. Author Summary: Studies of human perception and decision making have traditionally focused on scenarios where participants have to make estimates about continuous variables. However discrete variables are also common in our environment, potentially requiring different theoretical models. We describe ways to model such scenarios within the statistical framework of Bayesian inference and explain how aspects of such models can be teased apart experimentally. Using two experimental setups, a numerosity task and an area estimation task, we show that human participants do indeed rely on combinations of specific model components. Specifically we show that human learning in discrete tasks can be surprisingly fast and that participants can use the learned information in a way that is either optimal or near-optimal.