Combining Predictive Distributions

TL;DR

This paper explores methods for aggregating predictive distributions, analyzing their theoretical properties and practical performance across financial and weather forecasting applications.

Contribution

It introduces a unified prediction space framework for combining various types of predictive distributions and evaluates multiple aggregation formulas including linear and non-linear pools.

Findings

Generalized linear pools improve calibration.

Spread-adjusted pools effectively control dispersion.

Beta-transformed pools enhance forecast accuracy.

Abstract

Predictive distributions need to be aggregated when probabilistic forecasts are merged, or when expert opinions expressed in terms of probability distributions are fused. We take a prediction space approach that applies to discrete, mixed discrete-continuous and continuous predictive distributions alike, and study combination formulas for cumulative distribution functions from the perspectives of coherence, probabilistic and conditional calibration, and dispersion. Both linear and non-linear aggregation methods are investigated, including generalized, spread-adjusted and beta-transformed linear pools. The effects and techniques are demonstrated theoretically, in simulation examples, and in case studies on density forecasts for S&P 500 returns and daily maximum temperature at Seattle-Tacoma Airport.