Predictions as Statements and Decisions

Prediction is a complex notion, and different predictors (such as people, computer programs, and probabilistic theories) can pursue very different goals. In this paper I will review some popular kinds of prediction and argue that the theory of competitive on-line learning can benefit from the kinds of prediction that are now foreign to it. The standard goal for predictor in learning theory is to incur a small loss for a given loss function measuring the discrepancy between the predictions and the actual outcomes. Competitive on-line learning concentrates on a “relative” version of this goal: the predictor is to perform almost as well as the best strategies in a given benchmark class of prediction strategies. Such predictions can be interpreted as decisions made by a “small” decision maker (i.e., one whose decisions do not affect the future outcomes). Predictions, or probability forecasts, considered in the foundations of probability are statements rather than decisions; the loss function is replaced by a procedure for testing the forecasts. The two main approaches to the foundations of probability are measure-theoretic (as formulated by Kolmogorov) and gametheoretic (as developed by von Mises and Ville); the former is now dominant in mathematical probability theory, but the latter appears to be better adapted for uses in learning theory discussed in this paper. An important achievement of Kolmogorov’s school of the foundations of probability was construction of a universal testing procedure and realization (Levin, 1976) that there exists a forecasting strategy that produces ideal forecasts. Levin’s ideal forecasting strategy, however, is not computable. Its more practical versions can be obtained from the results of game-theoretic probability theory. For a wide class of forecasting protocols, it can be shown that for any computable game-theoretic law of probability there exists a computable forecasting strategy that produces ideal forecasts, as far as this law of probability is concerned. Choosing suitable laws of probability we can ensure that the forecasts agree with reality in requisite ways. Probability forecasts that are known to agree with reality can be used for making good decisions: the most straightforward procedure is to select decisions that are optimal under the forecasts (the principle of minimum expected loss). This gives, inter alia, a powerful tool for competitive on-line learning; I will describe its use for designing prediction algorithms that satisfy the property of universal consistency and its more practical versions. In conclusion of the paper I will discuss some limitations of competitive on-line learning and possible directions of further research.