Prediction with expert advice for the Brier game

We show that the Brier game of prediction is mixable and find the optimal learning rate and substitution function for it. The resulting prediction algorithm is applied to predict results of football and tennis matches. The theoretical performance guarantee turns out to be rather tight on these data sets, especially in the case of the more extensive tennis data.Comment: 34 pages, 22 figures, 2 tables. The conference version (8 pages) is published in the ICML 2008 Proceeding

Accuracy, Certainty and Surprise - A Prediction Market on the Outcome of the 2002 FIFA World Cup

In this chapter, we present our empirical investigation of the forecasting accuracy of a prediction market experiment drawn on the outcome of the World Cup 2002. We analyse the predictive accuracy of 64 markets and compare to bookmakers’ quotes and chance as benchmarks. We revisit the evaluation of Schmidt and Werwatz (Chapter 16) and compare our results directly to their findings. In addition, we propose a new method for testing predictive accuracy by means of a non-parametric test for the similarity of probability distributions and we evaluate the incorporation of information in market prices by comparing pre-match and half-time price data. We find a reversed favourite-longshot bias when analysing market prices before the start of the match and this bias does not disappear with the inflow of new information until half-time. Unlike the market based predictions bookmakers appear to be perfectly calibrated. Since there were substantial deviations in outcome between the 2000 European Championship and our data, we offer possible explanations for the much worse performance of the 2002 World Cup prediction market. Consistent with Schmidt and Werwatz (Chapter 16) prediction markets do assign relatively higher probabilities to the favourite when compared to the odds-setters. Together with a long streak of surprising outcomes this fact appears most likely to be responsible for the predictive inaccuracy.

Cascading Randomized Weighted Majority: A New Online Ensemble Learning Algorithm

With the increasing volume of data in the world, the best approach for learning from this data is to exploit an online learning algorithm. Online ensemble methods are online algorithms which take advantage of an ensemble of classifiers to predict labels of data. Prediction with expert advice is a well-studied problem in the online ensemble learning literature. The Weighted Majority algorithm and the randomized weighted majority (RWM) are the most well-known solutions to this problem, aiming to converge to the best expert. Since among some expert, the best one does not necessarily have the minimum error in all regions of data space, defining specific regions and converging to the best expert in each of these regions will lead to a better result. In this paper, we aim to resolve this defect of RWM algorithms by proposing a novel online ensemble algorithm to the problem of prediction with expert advice. We propose a cascading version of RWM to achieve not only better experimental results but also a better error bound for sufficiently large datasets.Comment: 15 pages, 3 figure

Jeffreys's law for general games of prediction: in search of a theory

We are interested in the following version of Jeffreys's law: if two predictors are predicting the same sequence of events and either is doing a satisfactory job, they will make similar predictions in the long run. We give a classification of instances of Jeffreys's law, illustrated with examples.Comment: 12 page