Calibrated Forecasts: The Minimax Proof

Consider a weather forecaster who announces each day a probability p that there will be rain tomorrow. The forecaster is said to be calibrated if, for each forecast p that is used, the relative frequency of rainy days out of those days in which the forecast was p is equal to p in the long run. The surprising result of Foster and Vohra (1998) is that calibration can be guaranteed, no matter what the weather will be. There are various proofs of this result, and there is a large literature on calibration and its uses; see the survey of Olszewski (2015) and the more recent paper of Foster and Hart (2021). A simple proof of the existence of calibrated forecasts, based on the minimax theorem, was provided by the author in 1995.