Bayesian Context Trees: Modelling and Exact Inference for Discrete Time Series

Abstract

We develop a new Bayesian modelling framework for the class of higher‐order, variable‐memory Markov chains, and introduce an associated collection of methodological tools for exact inference with discrete time series. We show that a version of the context tree weighting alg‐orithm can compute the prior predictive likelihood exa‐ctly (averaged over both models and parameters), and two related algorithms are introduced, which identify the a posteriori most likely models and compute their exact posterior probabilities. All three algorithms are deterministic and have linear‐time complexity. A family of variable‐dimension Markov chain Monte Carlo samplers is also provided, facilitating further exploration of the posterior. The performance of the proposed methods in model selection, Markov order estimation and prediction is illustrated through simulation experiments and real‐world applications with data from finance, genetics, neuroscience and animal communication. The associated algorithms are implemented in the R package BCT.

Context-tree weighting and Bayesian Context Trees: Asymptotic and non-asymptotic justifications

“…Here we collect the necessary definitions, assumptions, and basic results that will be used throughout the paper. All the results of this section (except for the straightforward observations in Proposition 2.2, proved in Section 4) can be found, along with more extensive discussion and details, in [21].…”

“…The statistical tools provided by the BCT framework have been found to provide efficient methods for very effective inference in a variety of applications [21,33,24,30,34]. In terms of the underlying theory, the Bayesian perspective adopted in [21] and this work is neither purely subjective, interpreting the prior and posterior as subjective descriptions of uncertainty pre-and post-data, respectively, nor purely objective, treating the resulting methods as simple black-box procedures [9].…”

“…The statistical tools provided by the BCT framework have been found to provide efficient methods for very effective inference in a variety of applications [21,33,24,30,34]. In terms of the underlying theory, the Bayesian perspective adopted in [21] and this work is neither purely subjective, interpreting the prior and posterior as subjective descriptions of uncertainty pre-and post-data, respectively, nor purely objective, treating the resulting methods as simple black-box procedures [9]. For example, we think of the MAP model as the most accurate, data-driven representation of the regularities present in a given time series, but we inform our analysis of the resulting inferential procedures by simulation experiments on hypothetical models, and by examining their frequentist properties; see, e.g., [3,Chapter 6] or [13,Chapter 4] for broad discussions of the relationship between the Bayesian and classical outlook.…”

“…At the same time as its importance for data compression was being recognised, it was also gradually becoming clear that the CTW algorithm can be best understood once it is properly rooted on a firm statistical foundation. Following a number of early relevant works [46,56,57,51,29], progress along this direction culminated in the recent development [21] of the Bayesian Context Trees (BCT) framework.…”

“…The BCT framework provides a general Bayesian foundation within which the CTW algorithm and its associated tools and techniques can be explored in a systematic and principled fashion. Indeed, in [21] it was shown that the CTW algorithm along with two generalisations of the context-tree maximising algorithm [57,51] can be used very effectively for Bayesian inference with discrete time series, in particular for model selection and prediction. A Markov chain Monte Carlo sampler was also developed for the posterior distribution over both models and parameters, and a more efficient, simple Monte Carlo sampler was introduced in [32,33], where the application of the BCT framework to estimation problems was explored further.…”