  Sequential data naturally arises from user engagement on digital platforms
like social media, music streaming services, and web navigation, encapsulating
evolving user preferences and behaviors through continuous information streams.
A notable unresolved query in stochastic processes is learning mixtures of
continuous-time Markov chains (CTMCs). While there is progress in learning
mixtures of discrete-time Markov chains with recovery guarantees
[GKV16,ST23,KTT2023], the continuous scenario uncovers unique unexplored
challenges. The intrigue in CTMC mixtures stems from their potential to model
intricate continuous-time stochastic processes prevalent in various fields
including social media, finance, and biology.
  In this study, we introduce a novel framework for exploring CTMCs,
emphasizing the influence of observed trails' length and mixture parameters on
problem regimes, which demands specific algorithms. Through thorough
experimentation, we examine the impact of discretizing continuous-time trails
on the learnability of the continuous-time mixture, given that these processes
are often observed via discrete, resource-demanding observations. Our
comparative analysis with leading methods explores sample complexity and the
trade-off between the number of trails and their lengths, offering crucial
insights for method selection in different problem instances. We apply our
algorithms on an extensive collection of Lastfm's user-generated trails
spanning three years, demonstrating the capability of our algorithms to
differentiate diverse user preferences. We pioneer the use of CTMC mixtures on
a basketball passing dataset to unveil intricate offensive tactics of NBA
teams. This underscores the pragmatic utility and versatility of our proposed
framework. All results presented in this study are replicable, and we provide
the implementations to facilitate reproducibility.
