MCMC analysis for a continuous time hidden Markov autoregressive process in disease progression

Abstract

A Markov Chain Monte Carlo method (MCMC) is adopted to estimate a continuous hidden Markov autoregressive model; best fitted to formulate the hidden behavior of a disease progression. The motivation behind this work is to unravel the latent disease stages during its development in some pathologies such as cancer, where during treatment the disease stage is not known directly unless we observe other parameters like bio-markers. By the way, we suppose we have individual patient observations (bio-markers'measurements) with different number of observation points and with non equidistant time intervals, hence we work in a continuous time framework. Usually, the likelihood is unavailable in analytic form for most of the Markov switching models (MSM); particularly that the hidden states are unknown parameters and by the way the likelihood is augmented with the hidden states as new parameters. To better deal with the augmented parameters and to use a likelihood up to constant, we call for the Bayesian MCMC tools by using data augmentation in the likelihood; which would alternate between estimating the parameters and evaluating the hidden states. The posterior densities for the parameters will be developed after prior specifications, and the hidden states will be computed by Gibbs Sampler block update using a forward filtering backward smoothing (FFBS) scheme. Finally, the estimation of transition rates between states is well explained as well as the MCMC algorithm results are provided.