Bayesian inference for discretely observed continuous time multi-state models

Multi-state models are frequently applied to represent processes evolving through a discrete set of states. Important classes of multi-state models arise when transitions between states may depend on the time passed since entry into the current state or on the time elapsed from the start of the process. The former models are called semi-Markov while the latter are known as inhomogeneous Markov models. Inference for both the models presents computational difficulties when the process is only observed at discrete time points with no additional information about the state transitions. In fact, in both the cases, the likelihood function is not available in closed form. To obtain Bayesian inference under these two classes of models, we reconstruct the entire unobserved trajectories conditioned on the observed points via a Metropolis-Hastings algorithm. As proposal density we use that given by the nested Markov models whose conditioned trajectories can easily be drawn with the uniformization technique. The resulting inference is illustrated via simulation studies and the analysis of two benchmark datasets for multi-state models.

Automated summary

Multi-state models are commonly used to represent processes that evolve through a discrete set of states. Two important classes of such models are semi-Markov models, where state transitions may depend on the time spent in the current state, and inhomogeneous Markov models, where transitions are dependent on the time elapsed from the start of the process. Inference for these models becomes computationally challenging when the process is observed only at discrete time points without additional information about state transitions. To address this, a Metropolis-Hastings algorithm is used to reconstruct the unobserved trajectories conditioned on the observed points. The resulting Bayesian inference is illustrated using simulation studies and analysis of benchmark datasets for multi-state models.