Supermartingales In Prediction With Expert Advice

The paper applies the method of defensive forecasting, based on the use of game-theoretic supermartingales, to prediction with expert advice. In the traditional setting of a countable number of experts and a finite number of outcomes, the Defensive Forecasting Algorithm is very close to the well-known Aggregating Algorithm both for mixable and non-mixable games. Not only the performance guarantees but also the predictions are the same for these two methods of fundamentally different nature. The paper also discusses a new setting where the experts can give advice conditional on the learnerís future decision, the setting of second-guessing experts. Both the algorithms can be adapted to the new setting and give the same performance guarantees as in the traditional setting. Finally, an application of defensive forecasting to a setting with several loss functions is outlined.