Variational Markov chain mixtures with automatic component selection

Markov state modeling has gained popularity in various scientific fields since it reduces complex time-series data sets into transitions between a few states. Yet common Markov state modeling frameworks assume a single Markov chain describes the data, so they suffer from an inability to discern heterogeneities. As an alternative, this paper models time-series data using a mixture of Markov chains, and it automatically determines the number of mixture components using the variational expectation-maximization algorithm.Variational EM simultaneously identifies the number of Markov chains and the dynamics of each chain without expensive model comparisons or posterior sampling. As a theoretical contribution, this paper identifies the natural limits of Markov state mixture modeling by proving a lower bound on the classification error. It then presents numerical experiments where variational EM achieves performance consistent with the theoretically optimal error scaling. The experiments are based on synthetic and observational data sets including Last.fm music listening, ultramarathon running, and gene expression. In each of the three data sets, variational EM leads to the identification of meaningful heterogeneities.