The Bayesian Context Trees (BCT) framework is a recently introduced, general collection of statistical and algorithmic tools for modelling, analysis and inference with discrete-valued time series. The foundation of this development is built in part on some well-known information-theoretic ideas and techniques, including Rissanen's tree sources and Willems et al.'s context-tree weighting algorithm. This paper presents a collection of theoretical results that provide mathematical justifications and further insight into the BCT modelling framework and the associated practical tools. It is shown that the BCT prior predictive likelihood (the probability of a time series of observations averaged over all models and parameters) is both pointwise and minimax optimal, in agreement with the MDL principle and the BIC criterion. The posterior distribution is shown to be asymptotically consistent with probability one (over both models and parameters), and asymptotically Gaussian (over the parameters). And the posterior predictive distribution is also shown to be asymptotically consistent with probability one.

翻译：贝叶斯上下文树（BCT）框架是近期提出的一套通用的统计与算法工具集合，用于离散值时间序列的建模、分析与推断。该框架的部分基础建立在若干著名信息论思想与技术之上，包括Rissanen的树信源和Willems等人的上下文树加权算法。本文提出一系列理论结果，为BCT建模框架及相关实用工具提供数学依据和更深入的理解。研究表明，BCT先验预测似然（对所有模型和参数取平均的时间序列观测概率）在逐点与极小化极大意义下均达到最优，与MDL原则和BIC准则一致。后验分布被证明在概率意义下渐近一致（对模型和参数均成立），且在参数上渐近服从高斯分布。后验预测分布同样被证明在概率意义下渐近一致。