Approximate Bayesian inference for discretely observed continuous‐time multi‐state models

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, 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 when the likelihood calculation is computationally too costly. In this article we investigate the potentiality of ABC techniques for multi‐state models both to obtain the posterior distributions of the model parameters and to compare Markov and semi‐Markov models. In addition, we will also exploit ABC methods to estimate and compare hidden Markov and semi‐Markov models when observed states are subject to classification errors. We illustrate the performance of the ABC methodology both with simulated data and with a real data example.