  This study introduces a novel approach for learning mixtures of Markov
chains, a critical process applicable to various fields, including healthcare
and the analysis of web users. Existing research has identified a clear divide
in methodologies for learning mixtures of discrete and continuous-time Markov
chains, while the latter presents additional complexities for recovery accuracy
and efficiency.
  We introduce a unifying strategy for learning mixtures of discrete and
continuous-time Markov chains, focusing on hitting times, which are well
defined for both types. Specifically, we design a reconstruction algorithm that
outputs a mixture which accurately reflects the estimated hitting times and
demonstrates resilience to noise. We introduce an efficient gradient-descent
approach, specifically tailored to manage the computational complexity and
non-symmetric characteristics inherent in the calculation of hitting time
derivatives. Our approach is also of significant interest when applied to a
single Markov chain, thus extending the methodologies previously established by
Hoskins et al. and Wittmann et al. We complement our theoretical work with
experiments conducted on synthetic and real-world datasets, providing a
comprehensive evaluation of our methodology.
