Optimal sequential probability assignment for individual sequences

Abstract

We compare the probabilities assigned to individual sequences by any sequential scheme, with the performance of the best 'batch' scheme in some class. For the class of finite-state (FS0 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. The bound and its sequential achievability establish a completely deterministic significance to the concept of predictive MDL.