Bayesian models of expert forecasts

This dissertation presents a general Bayesian approach to the expert consultation problem. In this problem, a decision maker with a prior distribution of a general random quantity of interest (e.g., a random variable or vector) consults a group of experts for their forecasts of the same random quantity of interest. These experts' forecasts are data for which a decision maker assigns a likelihood---a regular conditional distribution of experts' random distributions given the random quantity of interest itself. Importantly, we do not restrict experts' probability forecasts to come from any particular parametric family. After the experts report their forecasts or aspects thereof but prior to observing the random quantity of interest, the decision maker finds her posterior distribution of the random quantity of interest given the experts' reported forecasts through Bayesian updating.; In particular, we generalize previous Bayesian approaches to this problem in the following ways: (i) We propose a broader class of tractable likelihoods of nonparametric probability forecasts; (ii) We extend from the static, single-expert, univariate case to treat the dynamic, multi-expert, multivariate case; and (iii) We introduce a hierarchical model that parameterizes a decision maker's beliefs about each expert's forecasting ability.; Surprisingly, we find two limitations of our likelihood assignments in the continuous case: (i) They are proper likelihoods for a finite number of reported cumulative probabilities but are not proper likelihoods for approaching entire distribution functions; and (ii) The finite-dimensional likelihood of a random quantile function induced by our likelihood of a random distribution function is a mixture of singular and absolutely continuous pieces.; Finally, we introduce the notion of a dynamic random conditional distribution within a hierarchical model of a priori exchangeable experts dynamically forecasting stock index prices. In this application, we find posterior distributions of the parameters in the model and the final day's stock index price given past prices and forecasts, thus enabling the decision maker to learn jointly about experts' forecasting abilities and a new quantity interest.