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)框架是一种最近引入的,用于对离散值时间序列进行建模、分析和推断的一般性的统计和算法工具集。该发展的基础部分建立在一些著名的信息理论思想和技术之上,包括 Rissanen 的树源和 Willems 等人的上下文树加权算法。本文提供了一系列理论结果,为 BCT 建模框架和相关实用工具提供数学正当性和进一步的洞察力。结果表明,BCT 先验预测似然(在所有模型和参数的平均下,观察序列的概率)在点点和最小鞍点优化上是最优的,符合 MDL 原则和 BIC 标准。后验分布被证明在渐近成立的概率为1和渐近高斯上(在参数上)。后验预测分布也被证明在渐近成立的概率为1。