Bayesian inference for the Markov-modulated Poisson process with an outcome process

In medical research, understanding changes in outcome measurements is crucial for inferring shifts in health conditions. However, traditional methods often struggle with large, irregularly longitudinal data and fail to account for the tendency of individuals in poorer health to interact more frequently with the healthcare system. Additionally, clinical data can lack information on terminating events like death. To address these challenges, we start from the continuous-time hidden Markov model which models observed data as outcomes influenced by latent health states. Our extension incorporates a point process to account for the impact of health states on observation timings and includes a "death" state to model unobserved terminating events through a Poisson process, where transition rates depend on the latent health state. This approach captures both the severity of the disease and the timing of healthcare interactions. We present an exact Gibbs sampler procedure that alternates between sampling the latent health state paths and the model parameters. By including the "death" state, we mitigate biases in parameter estimation that would arise from solely modelling "live" health states. Simulation studies demonstrate that the proposed Gibbs sampler performs effectively. We apply our method to Canadian healthcare data, offering valuable insights for healthcare management.