# Strong Prequential Principle

We are interested in criteria of agreement between a sequence of forecasts $f_1,f_2,\ldots$ and a sequence of outcomes $x_1,x_2,\ldots$. The strong prequential principle says that:

• any such criterion should depend only on the actual observed sequences of forecasts $f_1,f_2,\ldots$ and outcomes $x_1,x_2,\ldots$, and not further on the strategies (if any) which might have produced these (the weak prequential principle), and
• there exists a criterion of agreement between a forecasting system $\phi$ and a sequence of outcomes $x_1,x_2,\ldots$ that has a stochastic justification and such that: for any forecasting system $\phi$ that would have produced $f_1,f_2,\ldots$ when fed with $x_1,x_2,\ldots$, the agreement between $f_1,f_2,\ldots$ and $x_1,x_2,\ldots$ is equivalent to the agreement between $\phi$ and $x_1,x_2,\ldots$.

Using criteria of agreement respecting the strong prequential principle makes it possible to use prequential models.

An alternative to the strong prequential principle, which is sometimes confused with the strong prequential principle itself, is: Suppose we have a criterion of agreement between a forecasting system and a sequence of realized outcomes, but we would like to have a criterion of agreement between a sequence of forecasts $f_1,f_2,\ldots$ and a sequence of outcomes $x_1,x_2,\ldots$, along the lines of the weak prequential principle. Then we could say that $f_1,f_2,\ldots$ and $x_1,x_2,\ldots$ agree with each other if and only if there is a forecasting system $\phi$ that: (1) outputs the forecasts $f_1,f_2,\ldots$ when fed with $x_1,x_2,\ldots$; (2) agrees with $x_1,x_2,\ldots$.

### Bibliography

• A. P. Dawid. Prequential analysis. In: Encyclopedia of Statistical Sciences, update volume 1, S. Kotz, Editor-in-Chief, pp. 464 - 470. Wiley, 1997. The strong prequential principle is first stated.
• A. P. Dawid and V. G. Vovk. Prequential probability: Principles and properties. Bernoulli 5:125 - 162, 1999. The strong and superstrong prequential principles are discussed.