WP 2020-11 Belief Distributions, Bayes Rule and Bayesian Overconfidence
AUTHORS: Glenn W. Harrison and J. Todd Swarthout
ABSTRACT: Previous tests of the consistency of beliefs with Bayes Rule tested only for bias, by comparing elicited subjective probabilities over binary events with the Bayesian posterior probability. We extend these tests by considering entire distributions, which allows us to evaluate not only bias but also dispersion of beliefs by individuals. Belief dispersion at the level of the individual can naturally be thought of as confidence. Whether subjects have too much or too little confidence can only be defined by the appropriate Bayesian confidence as defined by the variance of the posterior distribution. We show that individuals generally behave inconsistently with Bayes Rule defined over continuous events, and in very different ways over time. In the early stages of updating they exhibit too little confidence in their beliefs, and are also biased in terms of the mean of beliefs. But further updating leads rapidly to well-calibrated beliefs, both in terms of bias and confidence. However, continued updating leads to subjective belief distributions that are unbiased, but with significant overconfidence. Only by eliciting distributions are we able to assess the dispersion of an individual’s beliefs, giving us a natural metric for subjective confidence. The evidence involves the elicitation of beliefs with financial incentives, and tests Bayes Rule in a setting in which the posterior distribution is known for each individual and belief elicitation. In this setting, popular definitions of overconfidence in terms of “overestimation” or “overplacement” are correctly viewed as measures of bias.