Before revision: We examine and compare the predictive properties of classes of ensemble methods, including the recently developed Bayesian predictive synthesis (BPS). We develop a novel strategy based on stochastic processes, where the predictive processes are expressed as stochastic differential equations, evaluated using It\^{o show that a specific form of non-linear BPS (as in McAlinn and West, 2019) produces exact minimax predictive distributions for Kullback-Leibler risk and, under certain conditions, quadratic risk.
After revision: We analyze the combination of multiple predictive distributions for time series data when all forecasts are misspecified. We show that a specific form of non-linear BPS (as in McAlinn and West, 2019) produces exact minimax predictive distributions for Kullback-Leibler risk and, under certain conditions, quadratic risk.