An E cient Extension to Mixture Techniques forPrediction and Decision

We present an eecient method for maintaining mixtures of prunings of a prediction or decision tree that extends the previous methods for \node-based" prunings to the larger class of edge-based prunings. The method includes an online weight-allocation algorithm that can be used for prediction, compression and classiication. Although the set of edge-based prunings of a given tree is much larger than that of node-based prunings, our algorithm has similar space and time complexity to that of previous mixture algorithms for trees. Using the general online framework of Freund & Schapire (1997), we prove that our algorithm maintains correctly the mixture weights for edge-based prunings with any bounded loss function. We also give a similar algorithm for the logarithmic loss function with a corresponding weight-allocation algorithm. Finally, we describe experiments comparing node-based and edge-based mixture models for estimating the probability of the next word in English text, which show the advantages of edge-based models.