Published on Wed Nov 20 2019

Predictive properties and minimaxity of Bayesian predictive synthesis

Kōsaku Takanashi, Kenichiro McAlinn

We examine and compare the predictive properties of classes of ensemble methods, including the recently developed Bayesian predictive synthesis (BPS) We identify the conditions and mechanism for which non-linear synthesis improves over linear combinations; conditions that are commonly met in real world applications.

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Abstract

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}'s lemma. Using this strategy, we identify two main classes of ensemble methods: linear combination and non-linear synthesis, and show that a subclass of BPS is the latter. With regard to expected squared forecast error, we identify the conditions and mechanism for which non-linear synthesis improves over linear combinations; conditions that are commonly met in real world applications. We further 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. A finite sample simulation study is presented to illustrate our results.