Multivariate Markov Chain (MMC) models provide a powerful framework for
capturing dependencies among multiple interrelated processes that evolve
over time. They are widely used in finance, epidemiology, and genetics,
where understanding the joint behaviour of several variables is crucial
for prediction, risk assessment, and decision-making.

When all chains share the same discrete state space, a natural
parameterisation is through the joint transition probability matrix on
the extended state space formed by the Cartesian product of the marginal
spaces. However, the number of parameters grows rapidly with both the
number of states and the number of chains, creating major challenges for
estimation and inference.

In this talk, I will introduce alternative, general parameterisations
for MMC models that efficiently capture and characterise first-order
dependence between chains—such as conditional independence,
contemporaneous dependence, and Granger non-causality—while remaining
compatible with likelihood-based inference. I will also present a new
algorithm, the Generalized Individual Forward–Backward (GIFB) algorithm,
which provides exact (in a Monte Carlo sense) inference for latent
states in Hidden MMCs and reduces computational complexity from
exponential to quadratic in the number of chains.

The work is motivated by applications to multivariate Markov-modulated
Poisson processes for detecting disruption on the National Rail network
in Great Britain, using Twitter/X volume and content related to delays
and disturbances. If time permits, I will illustrate how the proposed
parameterisations can be applied to answer practically important
questions in this context.