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.
Highlights
People are often asked questions that require discrete numerical estimates
A numerosity task and an area estimation task, we show that human participants do rely on combinations of specific model components
Bayesian decision theory prescribes how to combine prior beliefs about states of the world, the likelihood that a state of the world generated an observation, and the decision function—the function which converts an uncertain representation into the response that maximizes expected reward
Summary
People are often asked questions that require discrete numerical estimates. When judging from a glance “How many people are in the room?” or “How many dots are on a screen?” the quantity to estimate is discrete and any sensible answer must be a whole number. Discrete numerical estimates are often required when the underlying quantity is continuous, very commonly when buying items. Bayesian decision theory prescribes how to combine prior beliefs about states of the world, the likelihood that a state of the world generated an observation, and the decision function—the function which converts an uncertain representation into the response that maximizes expected reward. The interaction between these components is fixed in Bayesian decision theory, but the prior, likelihood, and decision function can each take many forms. Identifying the prior characterizes how people represent past experience, identifying the likelihood indicates how people represent the evidential value of a new observation, and identifying the decision function characterizes how people convert uncertain beliefs into an estimate
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