On the Proofs of the Predictive Synthesis Formula

Abstract: Bayesian predictive synthesis is useful in synthesizing multiple predictive distributions coherently. However, the proof for the fundamental equation of the synthesized predictive density has been missing. In this technical report, we review the series of research on predictive synthesis, then fill the gap between the known results and the equation used in modern applications. We provide two proofs and clarify the structure of predictive synthesis.