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A hierarchical Bayesian framework is introduced for developing rich mixturemodels for real-valued time series, partly motivated by important applicationsin financial time series analysis. At the top level, meaningful discrete statesare identified as appropriately quantised values of some of the most recentsamples. These observable states are described as a discrete context-treemodel. At the bottom level, a different, arbitrary model for real-valued timeseries -- a base model -- is associated with each state. This defines a verygeneral framework that can be used in conjunction with any existing model classto build flexible and interpretable mixture models. We call this the BayesianContext Trees State Space Model, or the BCT-X framework. Efficient algorithmsare introduced that allow for effective, exact Bayesian inference and learningin this setting; in particular, the maximum a posteriori probability (MAP)context-tree model can be identified. These algorithms can be updatedsequentially, facilitating efficient online forecasting. The utility of thegeneral framework is illustrated in two particular instances: Whenautoregressive (AR) models are used as base models, resulting in a nonlinear ARmixture model, and when conditional heteroscedastic (ARCH) models are used,resulting in a mixture model that offers a powerful and systematic way ofmodelling the well-known volatility asymmetries in financial data. Inforecasting, the BCT-X methods are found to outperform state-of-the-arttechniques on simulated and real-world data, both in terms of accuracy andcomputational requirements. In modelling, the BCT-X structure finds naturalstructure present in the data. In particular, the BCT-ARCH model reveals anovel, important feature of stock market index data, in the form of an enhancedleverage effect.

The Bayesian Context Trees State Space Model for time series modelling and forecasting