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.
State-dependent regime switching diffusion processes or hybrid switching diffusion processes (HSD) are hard to simulate with classical methods which leads us to adopt an MCMC Bayesian approach very convenient to estimate complicated models such as the HSD one. In the HSD, the diffusion component is dependent on the switching discrete hidden regimes and the transition rates of the regime switching are dependent on the diffusion observations. Since in reality phenomena are only observed in discrete times, data imputation is called for to create more observations so as to have good approximations for the density of the diffusion process. Three categories of entities will be computed in a Bayesian context: The latent imputed observations, the regime switching states, and the parameters of the models. The latent imputed data is updated at random time intervals in block using a Metropolis Hastings algorithm. The switching states are computed by an adaptation of a forward filtering backward smoothing algorithm to a the HSD model. The parameters are estimated after prior specifications and conditional posterior densities formulation using Gibbs sampler or Metropolis Hastings algorithm.
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