Outcome-dependent density combination by mixture Bayesian predictive synthesis

In the problem of density combination, the decision maker receives multiple density functions and is tasked to learn which density is used most in which region of the outcome space, and to combine them into a single density. The outcome-dependent density combination can be cast as the Bayesian predictive synthesis, where the decision maker's uncertainty is naturally integrated. The models for the outcome-dependent weights have been considered in Johnson and West (2025), but limited to the Gaussian density kernel. In this study, we point out the limitation of the model flexibility of the Gaussian weights, and theoretically and numerically verify that the estimated weights can be least outcome-dependent. As an alternative, we propose the use of the softmax (logistic-normal) functions as the models for the weights, together with the bias-adjustment to the submitted densities. The posterior computation is feasible by using the custom Metropolis-Hastings method with the Pólya-gamma augmentation. The utility of the proposed method is illustrated in the real data analysis of movie theatre revenues, where a human expert reports the point quantile forecasts without any uncertainty quantification.