Recovering subjective probability distributions: a Bayesian approach

Abstract An individual reports subjective beliefs over continuous events using a proper scoring rule, such as the Quadratic Scoring Rule. Under mild additional assumptions, it is known that these reports reflect latent subjective beliefs if the individual is risk neutral. We demonstrate how to fully recover latent subjective beliefs if the individual is known to distort probabilities into decision weights using Rank Dependent Utility theory, and generalize all results for the complete class of proper scoring rules. As an immediate corollary, we can then exactly recover latent subjective beliefs using Expected Utility Theory. Using Bayesian econometric methods, we demonstrate how to recover belief distributions for individuals, and illustrate these results by recovering belief distributions about the “competitiveness” of individuals and the addictiveness of smoking. These results significantly widen the domain of applicability of proper scoring rules for eliciting latent subjective belief distributions.