The Bayesian Context Tree (BCT) framework is a recently introduced general collection of statistical and algorithmic tools for modeling, analysis, and inference using discrete-value time series. Part of the basis for this development is based on well-known information theory ideas and techniques such as Rissanen’s tree source and Willems et al.’s context tree weighting algorithm. This white paper presents a set of theoretical results that provide mathematical justification and further insight into the BCT modeling framework and related practical tools. The BCT prior prediction likelihood (probability of the observed time series averaged over all models and parameters) is expected to be pointwise and minimax-optimal, consistent with the MDL principle and the BIC criterion. shown. The posterior distribution is shown to be asymptotically consistent with probability 1 (in both model and parameters) and asymptotically Gaussian (in parameters). It also shows that the posterior predictive distribution agrees asymptotically with probability 1.
