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One of the main obstacles in the development of effective algorithms for inference and learning from discrete time series data, is the difficulty encountered in the identification of useful temporal structure. We will discuss a class of novel methodological tools for effective Bayesian inference and model selection for general discrete time series, which offer promising results on both small and big data. 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 resulting posterior and predictive distributions without resorting to MCMC. We have applied the resulting tools to numerous application-specific tasks, including on-line prediction, segmentation, classification, anomaly detection, entropy estimation, and causality testing, on data sets from different areas of application, including data compression, neuroscience, finance, genetics, and animal communication. Results on both simulated and real data will be presented.