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.

该研究介绍了一种新颖的连续时间马尔可夫链混合模型，重点研究了观察路径长度和混合参数对问题范畴的影响，并通过实验证明了离散化连续时间路径对混合模型的可学习性有重要影响，为不同问题情境下的算法选择提供了关键见解。

Markovletics: 学习连续时间马尔可夫链混合模型的方法及一种新应用

Markovletics: Methods and A Novel Application for Learning  Continuous-Time Markov Chain Mixtures

Many problems of practical interest rely on Continuous-time Markov
chains~(CTMCs) defined over combinatorial state spaces, rendering the
computation of transition probabilities, and hence probabilistic inference,
difficult or impossible with existing methods. For problems with countably
infinite states, where classical methods such as matrix exponentiation are not
applicable, the main alternative has been particle Markov chain Monte Carlo
methods imputing both the holding times and sequences of visited states. We
propose a particle-based Monte Carlo approach where the holding times are
marginalized analytically. We demonstrate that in a range of realistic
inferential setups, our scheme dramatically reduces the variance of the Monte
Carlo approximation and yields more accurate parameter posterior approximations
given a fixed computational budget. These experiments are performed on both
synthetic and real datasets, drawing from two important examples of CTMCs
having combinatorial state spaces: string-valued mutation models in
phylogenetics and nucleic acid folding pathways.

本研究提出了一种基于粒子的蒙特卡罗方法，其中持续时间通过解析方法边际化，最终在具有组合状态空间的连续时间马尔可夫链上实现更准确的参数后验近似。