Bayesian Estimation of Transition Rates in Two-State Nonhomogeneous Markov Jump Processes With Intermittent Observations: An Honest-Time Data-Augmentation Approach

Abstract

A possibly time-dependent transition intensity matrix or generator (𝑄⁡(𝑡)) characterizes the law of a Markov jump process (MP). For a time-homogeneous MP, the transition probability matrix (TPM) can be expressed as a matrix exponential of 𝑄 . However, when dealing with a time nonhomogeneous MP, there is often no simple analytical form of the TPM in terms of 𝑄⁡(𝑡) , unless all the 𝑄⁡(𝑡) commute. This poses a challenge because when a continuous MP is observed intermittently, a TPM is required to build a likelihood. In this paper, we show that the Bayesian estimation of the transition intensities of a two-state nonhomogeneous Markov model can be carried out by augmenting the intermittent observations with honest random times associated with two independent driving Poisson point processes, and that sampling the full path is not required. We thus propose a Bayesian data augmentation algorithm wherein the observations are augmented to include honest times, thus facilitating the sampling of the model parameters. Finally, we illustrate our approach by simulating a continuous MP and by using observed (intermittent) time grids extracted from real clinical visits data.