ASYMPTOTIC CALIBRATION

Can we forecast the probability of an arbitrary sequence of events hap pening so that the stated probability of an event happening is close t o its empirical probability? We can view this prediction problem as a game played against Nature, where at the beginning of the game Nature picks a data sequence and the forecaster picks a forecasting algorithm . If the forecaster is not allowed to randomise, then Nature wins; the re will always be data for which the forecaster does poorly. This pape r shows that, if the forecaster can randomise, the forecaster wins in the sense that the forecasted probabilities and the empirical probabil ities can be made arbitrarily close to each other.