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）框架是近期引入的一套用于离散值时间序列建模、分析与推断的通用统计与算法工具。该框架的基础部分源于若干著名的信息论思想与技术，包括Rissanen的树源模型以及Willems等人的上下文树加权算法。本文提出了一系列理论成果，为BCT建模框架及其相关实用工具提供了数学论证与更深层次的见解。研究表明，BCT先验预测似然（对所有模型与参数取平均的时间序列观测概率）在逐点意义和极小化极大意义下均为最优，符合MDL原则与BIC准则。后验分布在模型与参数层面均以概率1渐近一致，且参数层面具有渐近正态性。后验预测分布同样以概率1渐近一致。