Efficiently Encoding and Modeling Subjective Probability Distributions for Quantitative Variables

Expert forecasts of quantitative variables in the form of continuous subjective probability distributions are more useful to decision makers than are point estimates or confidence intervals. We present 2 experiments using participants recruited via the Internet aimed at (a) developing methods for estimating and modeling continuous subjective distributions from small numbers of judgments, and (b) assessing the effects of procedural variables on forecasting accuracy and difficulty. Experiment 1 assessed the feasibility of the proposed methods by having participants provide specified quantiles for ratios of area of geometric figures. Gamma and Weibull distributions fit the judgments very well and yielded mean and variance estimates that matched those obtained via established nonparametric methods. In Experiment 2, participants forecasted 3 future values: the date of an Apple product release announcement, the proportion of 2012 Summer Olympics medals that the United States and China would win, and the high temperature in their locality exactly 2 weeks hence. Between-participants variables were number of cut points (3 or 5) and response format (quantiles, cumulative probabilities, or interval probabilities). Overall, probability estimates were better than quantile estimates in terms of accuracy and ease of responding. Five cut points took longer than 3, but did not systematically improve accuracy. Gamma distributions fit the date forecasts well, normal distributions fit the temperature forecasts well, and beta distributions fit the proportion forecasts well. The results are very encouraging for rapid and efficient encoding and modeling of probabilistic forecasts of quantitative variables.