Decision-focused linear pooling for probabilistic forecast combination

In real-world settings, decision-makers often have access to multiple forecasts for the same unknown quantity. Combining different forecasts has long been known to improve forecast quality, as measured by scoring rules in the case of probabilistic forecasting. However, improved forecast quality does not always translate into better decisions in a downstream problem that utilizes the resultant combined forecast as input. To this end, this work proposes a novel probabilistic forecast combination approach that accounts for the downstream stochastic optimization problem by which the decisions will be made. We propose a linear pool of probabilistic forecasts where the respective weights are learned by minimizing the expected decision cost of the induced combination, which we formulate as a nested optimization problem. Two methods are proposed for its solution: a gradient-based method that utilizes differential optimization layers, and a performance-based weighting method. The proposed decision-focused combination approach is validated in two integral problems associated with renewable energy integration in low-carbon power systems and compared against well-established combination methods. Namely, we examine an electricity market trading problem under stochastic solar production and a grid scheduling problem under stochastic wind production. The results illustrate that the proposed approach leads to lower expected downstream costs, while optimizing for forecast quality when estimating linear pool weights does not always translate into better decisions. Notably, optimizing for a combination of downstream cost and an accuracy-oriented scoring rule consistently leads to better decisions while also improving forecast quality.