Reliable Decisions with Threshold Calibration

Decision makers rely on probabilistic forecasts to predict the loss of different decision rules before deployment. When the forecasted probabilities match the true frequencies, predicted losses will be accurate. Although perfect forecasts are typically impossible, probabilities can be calibrated to match the true frequencies on average. However, we find that this \textit{average} notion of calibration, which is typically used in practice, does not necessarily guarantee accurate decision loss prediction. Specifically in the regression setting, the loss of threshold decisions, which are decisions based on whether the forecasted outcome falls above or below a cutoff, might not be predicted accurately. We propose a stronger notion of calibration called threshold calibration, which is exactly the condition required to ensure that decision loss is predicted accurately for threshold decisions. We provide an efficient algorithm which takes an uncalibrated forecaster as input and provably outputs a threshold-calibrated forecaster. Our procedure allows downstream decision makers to confidently estimate the loss of any threshold decision under any threshold loss function. Empirically, threshold calibration improves decision loss prediction without compromising on the quality of the decisions in two real-world settings: hospital scheduling decisions and resource allocation decisions.