Bayesian Estimation of Multivariate Autoregressive Hidden Markov Model with Application to Breast Cancer Biomarker Modeling

Abstract

In this work, a first-order autoregressive hidden Markov model (AR(1)HMM) is proposed. It is one of the suitable models to characterize a marker of breast cancer disease progression essentially the progression that follows from a reaction to a treatment or caused by natural developments. The model supposes we have observations that increase or decrease with relation to a hidden phenomenon. We would like to discover if the information about those observations can let us learn about the progression of the phenomenon and permit us to evaluate the transition between its states (supposed discrete here). The hidden states governed by the Markovian process would be the disease stages, and the marker observations would be the depending observations. The parameters of the autoregressive model are selected at the first level according to a Markov process, and at the second level, the next observation is generated from a standard autoregressive model of first order (unlike other models considering the successive observations are independents). A Markov Chain Monte Carlo (MCMC) method is used for the parameter estimation, where we develop the posterior density for each parameter and we use a joint estimation of the hidden states or block update of the states.

An MCMC computational approach for a continuous time state-dependent regime switching diffusion process

“…Instead of simulating each hidden state separately, Chib [9] developed a method called block update of the hidden states in which we simulate the full latent data for each individual i, i = 1, 2, ...., N . Its algorithm was adapted to a multivariate autoregressive hidden Markov model in [30]. We will adapt this algorithm to the HSD process, with the modification that our model suppose observations with different lengths and non equidistant intervals.…”