Bayesian state combining for context models

The best-performing on-line methods for estimating probabilities of symbols in a sequence (required for computing minimal codes) use context trees with either information-theoretic state selection or context-tree weighting. This paper derives de novo from Bayes' theorem, a novel technique for modeling sequences on-line with context trees, which we call "Bayesian state combining" or BSC. BSC is comparable in function to both information-theoretic state selection and context-tree weighting. However, it is a truly distinct alternative to either of these techniques, which like BSC, can be viewed as "dispatchers" of probability estimates from the set of competing, memoryless models represented by the context tree. The resulting technique handles sequences over m-ary input alphabets for arbitrary m and may employ any probability estimator applicable to context models (e.g., Laplace, Krichevsky-Trofimov, blending, and more generally, mixtures). In experiments that control other (256-ary) context-tree model features such as Markov order and probability estimators, we compare the performance of BSC and information-theoretic state selection. The background notation and concepts are reviewed, as required to understand the modeling problem and application of our result. The leading notion of the paper is derived, which dynamically maps certain states in context-models to a set of mutually exclusive hypotheses and their prior and posterior probabilities. The efficient sequential computation of the posterior probabilities of the hypotheses, which was made possible via a non-obvious application of the percolating description-length update mechanism introduced by Bunton (see Proceedings Data Compression Conference, IEEE Computer Society Press, 1997) is described. The preliminary empirical performance of the technique on the Calgary Corpus is presented, the relationship of BSC to information-theoretic state selection and context-tree weighting is discussed.