Abstract:

Calibration means that for each forecast x the average of the realized actions in the periods in which the forecast was x is, in the long run, close to x. Calibration can always be guaranteed (Foster and Vohra 1998), but it requires the forecasting procedure to be stochastic. By contrast, smooth calibration, which combines in a continuous manner nearby forecasts, can be guaranteed by a deterministic procedure (Foster and Hart 2018). In the present paper we develop the concept of forecast-hedging, which consists of choosing the forecasts in such a way that, no matter what the realized action will be, the expected forecasting track record can only improve. This approach integrates the existing calibration results by obtaining them all from the same simple basic argument, and at the same time differentiates between them according to the forecast-hedging tools that are used: deterministic and fixed point-based vs. stochastic and minimax-based. Additional benefits are new calibration procedures in the one-dimensional case that are simpler than all known such procedures, and a short proof for deterministic smooth calibration, in contrast to the complicated existing proof.

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