The reproducible properties of correct forecasts.

*(English)*Zbl 1071.62084Summary: Each period one outcome out of finitely many possibilities is observed. Each period a forecaster announces some probability for the future outcomes based on the available data. An outsider wants to know if the forecaster has some knowledge of the data generating process. Let a test be an arbitrary function from sequences of forecasts and outcomes to \(\{0,1\}\). When the test returns a 0 the test is said to reject the forecasts based on the outcome sequence. When the test returns a 1 the test is said to not reject the forecasts based on the outcome sequence. Consider any test that does not reject the truth, i.e., it does not reject when the announced forecasts are the conditional probabilities of the data generating process. Based on Ky Fan’s [Proc. Natl. Acad. Sci. USA 39, 42–47 (1953; Zbl 0050.06501)]. Minimax theorem, I show that it is possible to produce forecasts that will not be rejected on any sequence of outcomes.