The Bayesian Context Trees State Space Model: Interpretable mixture models for time series

A general hierarchical Bayesian framework is introduced for mixture modelling of real-valued time series, including a collection of effective tools for learning and inference. At the top level, a discrete context (or `state') is extracted for each sample, consisting of a discretised version of some of the most recent observations preceding it. The set of all relevant contexts are represented as a discrete context tree. At the bottom level, a different real-valued time series model is associated with each context (i.e., with each state). This defines a very general framework that can be used in conjunction with any existing model class to build flexible and interpretable mixture models. We introduce algorithms that allow for efficient, exact Bayesian inference; in particular, the maximum a posteriori probability (MAP) model, including the relevant MAP context tree, can be identified exactly. These algorithms can be updated sequentially, facilitating efficient online forecasting. The utility of the general framework is illustrated in detail when autoregressive (AR) models are used at the bottom level, resulting in a nonlinear AR mixture model. Our methods are found to outperform several state-of-the-art techniques on both simulated and real-world data from economics and finance, both in terms of forecasting accuracy and computational requirements.