On Sequence Prediction for Arbitrary Measures

Daniil Ryabko, Marcus Hutter

Abstract: Suppose we are given two probability measures on the set of one-way infinite finite-alphabet sequences and consider the question when one of the measures predicts the other, that is, when conditional probabilities converge (in a certain sense) when one of the measures is chosen to generate the sequence. This question may be considered a refinement of the problem of sequence prediction in its most general formulation: for a given class of probability measures, does there exist a measure which predicts all of the measures in the class? To address this problem, we find some conditions on local absolute continuity which are sufficient for prediction and which generalize several different notions which are known to be sufficient for prediction. We also formulate some open questions to outline a direction for finding the conditions on classes of measures for which prediction is possible.