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Calibration with many checking rules. (English) Zbl 1082.90544
Summary: Each period an outcome (out of finitely many possibilities) is observed. For simplicity assume two possible outcomes, $$a$$ and $$b$$. Each period, a forecaster announces the probability of an occurring next period based on the past.
Consider an arbitrary subsequence of periods (e.g., odd periods, even periods, all periods in which $$b$$ is observed, etc.). Given an integer $$n$$, divide any such subsequence into associated subsubsequences in which the forecast for a is between $$[i /n, i+1/n)$$, $$i \in \{0,1,\dots,n\}$$.
We compare the forecasts and the outcomes (realized next period) separately in each of these subsubsequences. Given any countable partition of $$[0,1]$$ and any countable collection of subsequences, we construct a forecasting scheme such that for all infinite strings of data, the long-run average forecast for a matches the long-run frequency of realized $$a$$’s.

##### MSC:
 90B99 Operations research and management science 91A35 Decision theory for games 62M20 Inference from stochastic processes and prediction
##### Keywords:
forecast, probabilistic forecast, calibration
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