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

TL;DR

This paper introduces a novel pseudo-marginal MCMC method for estimating transition rates in two-state non-homogeneous Markov jump processes using intermittent observations and honest times, avoiding full path sampling.

Contribution

It develops a new approach combining honest times and pseudo-marginal MCMC to estimate transition intensities without requiring continuous path data.

Findings

Effective estimation of transition rates from intermittent data.

Simulation and real clinical data demonstrate the method's practicality.

Avoids the need for full path sampling in non-homogeneous Markov models.

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.