CALIBRATED FORECASTING AND MERGING + by

Consider a general finite-state stochastic process governed by an unknown objective probability distribution. Observing the system, a forecaster assigns subjective probabilities to future states. The resulting subjective forecast merges to the objective distribution if, with time, the forecasted probabilities converge to the correct (but unknown) probabilities. The forecast is calibrated if observed long-run empirical distributions coincide with the forecasted probabilities. This paper links the unobserved reliability of forecasts to their observed empirical performance by demonstrating full equivalence between notions of merging and of calibration. It also indicates some implications of this equivalence for the literatures of forecasting and learning.