OPTIMAL SEQUENTIAL PROBABILITY ASSIGNMENT FOR INDIVIDUAL SEQUENCES

The problem of sequential probability assignment for individual sequences is investigated. We compare the probabilities assigned by any sequential scheme to the performance of the best ''batch'' scheme (model) in some class. For the class of finite-state schemes and other related families, we derive a deterministic performance bound, analogous to the classical (probabilistic) Minimum Description Length (MDL) bound. It holds for ''most'' sequences, similarly to the probabilistic setting, where the bound holds for ''most'' sources in a class. It is shown that the bound can be attained both pointwise and sequentially for any model family in the reference class and without any prior knowledge of its order. This is achieved by a universal scheme based on a mixing approach. The bound and its sequential achievability establish a completely deterministic significance to the concept of predictive MDL.