Semi-Markov processes are stochastic processes widely used to model many phenomena in finance, as well as in different disciplines. However, several improvements might be made in the Bayesian computational inference framework. The goal of this work is to provide a Markov chain Monte Carlo method for estimating parameters of multi state processeses, extending some estimators already provided in literature for Markov jump Processes, to the more general case of semi-Markov processes. Specifically, for discrete observation of continuous time processes, our algorithm will be able to simulate the trajectories between the observed point and estimate the parameters governing the process. This by a two-phase algorithm: in the first step the parameters generating the process are simulated, while in the second phase these are used for the rate matrix generation, which is input the function that simulates semi-Markov paths and provide information for the next iteration.
Our idea is to extend the Rao and Teh algorithm to the more general semi-Markov case. This in order to avoid the assumption that transitions between states do not depend on the time since entry into the current state. Indeed, a more general algorithm allows to get estimations in Markov and semi-Markov models, without any assumption of independence between transitions and state entrance time. For doing that, we need to assume as sojourn time distributions a generalization of the exponential: we choose a Weibull distribution, which for shape parameter equal to one is an exponential. Hence, the first step of the algorithm will have a further step with respect to the Rao and Teh one: starting from a rate matrix as input as before, the sojourn times produced by the uniformization function will be evaluated with a Metropolis Hastings step, having a Weibull as target distribution. As for Rao and Teh, the accepted paths will represent information for the posterior distribution of the parameters generating the process, also there simulated in the second phase: transitions between each state to the others are still multinomial distributions and provide posterior information for the conjugate dirichlet transition probabilities, while ¿ since sojourn times are Weibull ¿ there will be a further parameter to simulate. Inference for Weibull parameters will be made by using a Metropolis Hastings step, with bivariate Normal proposal, while right now, the idea is to use non-informative priors for each of the parameters. The values of the parameters simulated in the second step give a rate matrix used as input for the first step in the next iteration in this algorithm as well.