# Probability Forecasting

The probability forecasting is a forecasting of the probability of one or more events, or probability predictions. These events can be mutually exclusive or not. Probability forecasting is an important application of the on-line prediction. The game-theoretic probability includes probability forecasting as one of the base elements.

#### Algorithms

There are many algorithms which can be applied to make probability forecasts. One can use some assumptions about the data, and thus use Bayes-based techniques, or other statistical learning theory techniques for it. These techniques usually work well for empirical data, but their theoretical guarantees are strongly restricted by the assumptions of the nature of the data.

If we consider the on-line prediction task of one event, one can use Exponential weights algorithms to predict the value between 0 and 1 as a probability of this event. In this case the real outcomes will be 0 or 1 exactly. These algorithms can provide strong theoretical guarantees on their performance in the correspondence to the chosen loss function. The Defensive forecasting technique providing K29 algorithm automatically allows forecasts to be well-calibrated (see below). To provide forecasts for more than two mutually exclusive events given the expert advice, one can use the Strong Aggregating Algorithm as it is described in Vovk and Zhdanov (2008).

#### Quality measure

To measure the quality of probability forecasts one usually uses the loss function which is a proper scoring rule, like square-loss or logarithmic loss. The other ways to check the quality of forecasts is a calibration and resolution features. Calibration, for example, ensures that the given value of the forecast corresponds to a real probability of the event predicted. Read more about coherence and expertise in Winkler (1986).

#### Application areas

The probability forecasting is very useful for meteorology, finance, medical diagnosis, and many other areas. One can apply the probability forecasting for time-series prediction by predicting the number of interval of the change of the current sequence.

### Bibliography

- Vovk, Vladimir and Zhdanov, Fedor. Prediction with expert advice for the Brier game. In: Proceedings of the 25th International Conference on Machine Learning and arXiv technical report.
- Winkler, R. L. (1986). On good probability appraisers. In P. Goel & A. Zellner (Eds.), Bayesian inference and decision techniques (pp. 265–278). Amsterdam: Elsevier Science Publishers B.V