Beyond time-homogeneity for continuous-time multistate Markov models

Multistate Markov models are a canonical parametric approach for data mod- eling of observed or latent stochastic processes supported on a finite state space. Continuous-time Markov processes describe data that are observed irregularly over time, as is often the case in longitudinal medical and biological data sets, for exam- ple. Assuming that a continuous-time Markov process is time-homogeneous, a closed- form likelihood function can be derived from the Kolmogorov forward equations – a system of differential equations with a well-known matrix-exponential solution. Un- fortunately, however, the forward equations do not admit an analytical solution for continuous-time, time-inhomogeneous Markov processes, and so researchers and prac- titioners often make the simplifying assumption that the process is piecewise time- homogeneous. In this paper, we provide intuitions and illustrations of the potential biases for parameter estimation that may ensue in the more realistic scenario that the piecewise-homogeneous assumption is violated, and we advocate for a solution for likelihood computation in a truly time-inhomogeneous fashion. Particular focus is afforded to the context of multistate Markov models that allow for state label mis- classifications, which applies more broadly to hidden Markov models (HMMs), and Bayesian computations bypass the necessity for computationally demanding numeri- cal gradient approximations for obtaining maximum likelihood estimates (MLEs).