Combining and Extremizing Real-Valued Forecasts

The weighted average is by far the most popular approach to combining multiple forecasts of some future outcome. This paper shows that both for probability or real-valued forecasts, a non-trivial weighted average of different forecasts is always sub-optimal. More specifically, it is not consistent with any set of information about the future outcome even if the individual forecasts are. Furthermore, weighted averaging does not behave as if it collects information from the forecasters and hence needs to be extremized, that is, systematically transformed away from the marginal mean. This paper proposes a linear extremization technique for improving the weighted average of real-valued forecasts. The resulting more extreme version of the weighted average exhibits many properties of optimal aggregation. Both this and the sub-optimality of the weighted average are illustrated with simple examples involving synthetic and real-world data.