Enumerative coding for tree sources

Given a parametric model family, the set of sequences of length n over a finite alphabet A is partitioned into type classes, where two sequences belong to the same class if and only if they are assigned the same probability by all models in the family. Since all sequences in a class are equiprobable, the universal probability assignment problem for the given model family reduces to optimally assigning probabilities to type classes. This reduction is optimally performed by the Normalized Maximum Likelihood (NML) code, which can be interpreted as a description of the type, generated by assigning to it a probability proportional to its ML probability, followed by an enumeration of the sequences in the type class. Unfortunately, implementing the NML code is difficult even for the simplest model classes. Other universal methods, based, for example, on the Krichevskii-Trofimov sequential probability assignment, are computationally efficient and also assign the same code length to all the sequences of a given type. They do not, however, provide a separate and identifiable description of the type. In this paper, we are interested in universal enumerative source codes that possess both qualities: they provide a separate description of the type class of the encoded sequence, and this description can be efficiently computed. By "efficient computation" we mean one with running time that is polynomial in the length of the input sequence, as well as in the number of model parameters.