Bayesian Predictive Synthesis: Forecast Calibration and Combination
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The combination of forecast densities, whether they result from a set of models, a group of consulted experts, or other sources, is becoming increasingly important in the fields of economics, policy and finance, among others. Requiring methodology that goes beyond standard Bayesian model uncertainty and model mixing - with its well-known limitations based on a clearly proscribed theoretical basis - multiple 'density combination' methods have been proposed. While some proposals have demonstrated empirical success, most apparently lack a core philosophical and theoretical foundation. Interesting recent examples generalize the common 'linear opinion pool' with flexible mixing weights that depend on the forecast variable itself - i.e., outcome-dependent mixing. Taking a foundational subjective Bayesian perspective, we show that such a density combination scheme is in fact justified as one example of Bayesian agent opinion analysis, or 'predictive synthesis'. This logically coherent framework clearly delineates the underlying assumptions as well as the theoretical constraints and limitations of many combination 'rules', defining a broad class of Bayesian models for the general problem. A number of examples, including an application to a set of predictive densities in foreign exchange, provide illustrations.
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