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 algorithm can compute the prior predictive likelihood exactly (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.

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.

We revisit the statistical foundation of the celebrated context tree weighting (CTW) algorithm, and we develop a Bayesian modelling framework for the class of higher-order, variable-memory Markov chains, along with an associated collection of methodological tools for exact inference for discrete time series. In addition to deterministic algorithms that learn the a posteriori most likely models and compute their posterior probabilities, we introduce a family of variable-dimension Markov chain Monte Carlo samplers, 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.

The identification of useful temporal dependence structure in discrete time series data is an important component of algorithms applied to many tasks in statistical inference and machine learning, and used in a wide variety of problems across the spectrum of biological studies. Most of the early statistical approaches were ineffective in practice, because the amount of data required for reliable modelling grew exponentially with memory length. On the other hand, many of the more modern methodological approaches that make use of more flexible and parsimonious models result in algorithms that do not scale well and are computationally ineffective for larger data sets. In this paper we describe a class of novel methodological tools for effective Bayesian inference for general discrete time series, motivated primarily by questions regarding data originating from studies in genetics and neuroscience. Our starting point is the development of a rich class of Bayesian hierarchical models for variable-memory Markov chains. The particular prior structure we adopt makes it possible to design effective, linear-time algorithms that can compute most of the important features of the relevant posterior and predictive distributions without resorting to Markov chain Monte Carlo simulation. The origin of some of these algorithms can be traced to the family of Context Tree Weighting (CTW) algorithms developed for data compression since the mid-1990s. We have used the resulting methodological tools in numerous application-specific tasks (including prediction, segmentation, classification, anomaly detection, entropy estimation, and causality testing) on data from different areas of application. The results obtained compare quite favourably with those obtained using earlier approaches, such as Probabilistic Suffix Trees (PST), Variable-Length Markov Chains (VLMC), and the class of Markov Transition Distributions (MTD).