The parameters of a discrete stationary Markov model are transition
probabilities between states. Traditionally, data consists in sequences of
observed states for a given number of individuals over the whole observation
period. In such a case, the estimation of transition probabilities is
straightforwardly made by counting one-step moves from a given state to
another. In many real-life problems, however, the inference is much more
difficult as state sequences are not fully observed, namely the state of each
individual is known only for some given values of the time variable $t$. In
this paper we give a review of this field, focusing on Monte Carlo Markov Chain
(MCMC) algorithms to perform Bayesian inference and evaluate posterior
distributions of the transition probabilities in this missing-data framework.
We also propose a way to accelerate the classical Metropolis-Hastings technique
for typical reliability problems, taking advantage of the dependence between
the matrix rows to build an adaptive MCMC.