Inference for continuous time multi-state models presents considerable computational difficulties when the process is only observed at discrete time points with no additional information about the state transitions. In fact, for general multi-state Markov model, the evaluation of the likelihood function is possible only via intensive numerical approximations. Moreover, in real applications, transitions between states may depend on the time since entry into the current state and semi-Markov models, where the likelihood function is not available in closed form, should be fitted to the data.

Approximate Bayesian Computation (ABC) methods, which make use only of comparisons between simulated and observed summary statistics, represent a solution to intractable likelihood problems and provide alternative algorithms whenthe likelihood calculation is computationally too costly. In this paper we investigate the potentiality of ABC techniques for multi-state models by means of simulations and a real data example. Both the problems of parameter estimation and model choice will be considered.