Estimating Transition Rates in Two-State Non-Homogeneous Markov Jump Processes with Intermittent Observations: A Pseudo-Marginal McMC Approach via Honest Times

2025-07-22

AI Summary

Estimating time-varying transition rates for two-state nonhomogeneous Markov jump processes under intermittent observations remains challenging: the time-dependent generator $Q(t)$ precludes closed-form transition probabilities, hindering conventional data-augmented likelihood construction. To address this, we propose a Bayesian inference framework grounded in “honest random times”—latent event times coupled with an auxiliary Poisson process—thereby bypassing explicit computation of transition matrices and enabling tractable pseudo-marginal likelihood formulation. Our approach employs pseudo-marginal MCMC to draw posterior samples efficiently without full path augmentation. Extensive simulation studies and analysis of real-world clinical longitudinal data demonstrate robustness and accuracy under irregular, sparse observation schemes. The method provides an interpretable, computationally feasible tool for modeling time-varying hazards in medical longitudinal studies.

Technology Category

Reasoning under Uncertainty: Probabilistic ProgrammingMachine Learning: Probabilistic Circuits and Graphical ModelsIntelligent Robots: State Estimation

Application Category

User Modeling, Personalization and Recommendation: Studies of user behavior, including longitudinal effects of personalized systemsGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 non-homogeneous MP, there is often no simple analytical form of the TPM in terms of $Q(t)$, unless they all 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 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 propose a pseudo-marginal McMC algorithm to estimate the transition rates using the augmented data. Finally, we illustrate our approach by simulating a continuous MP and by using observed (intermittent) time grids extracted from real clinical visits data.

Problem

• Estimating transition rates in non-homogeneous Markov jump processes
• Handling intermittent observations without full path sampling
• Developing pseudo-marginal McMC for time-dependent transition intensities

Innovation

• Augmenting observations with honest random times
• Using pseudo-marginal McMC for transition rates
• Avoiding full path sampling in estimation