Parametrisations for inference on the dependency structures in Multivariate Markov Chains
Publication Date
July 30, 2026
Creator
Jiale Tao
Abstract
Multivariate Markov Chains (MMCs) provide a powerful framework for capturing dependencies across multiple interrelated processes that
evolve over time. In literature, the natural way to represent MMC dynamics is through a joint transition matrix on the extended state space formed by the Cartesian product of all marginal state spaces. However, such a joint transition matrix by the Cartesian product does not explicitly reveal the underlying local dependency structures—for example, whether one group of chains depends on another, or whether interactive effects exist between groups.
To address this, this thesis develops two novel reparametrisations, the Univariate Marginal Model and the Hierarchical Marginal Model, which re-express the joint transition matrix in terms of marginal probabilities. These probabilities are further decomposed via multinomial logistic regression, yielding regression coefficients that not only provide clear interpretations of dependency structures but also allow explicit encoding and detection of these dependency structures such as conditional independence, contemporaneous independence, and Granger non-causality.
In many practical applications, the underlying MMCs are unobserved, giving rise to Hidden Markov Models (HMMs) where the hidden dynamics have to be inferred from the emitted observations. Classical methods such as the Forward–Backward algorithm become computationally intractable as the number of chains grows. To solve this, we develop the Generalised Individual Forward–Backward (GIFB) algorithm, an exact (in Monte Carlo sense) Gibbs sampler that updates hidden states group-wise, reducing computational cost from exponential to quadratic in the number of chains. Since GIFB operates on marginal transition probabilities, it naturally integrates with our earlier marginal models, enabling efficient full inference in Multivariate Hidden Markov Models (MHMMs). Finally, we applied this framework to real Twitter/X data in the context of UK railways to detect dependency structures and disruptions, demonstrating its effectiveness in practice.
Item Type
ethesis
Thesis Type
PhD
Supervisors
University of Nottingham
University of Nottingham
Subjects (LC)
Associated Schools / Departments
UoN Repository URI
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