An independent mixture model will not do for the earthquake series because − by definition − it does not allow for the serial dependence in the observations. The sample autocorrelation function (ACF), clearly indicates that the observations are serially dependent. One way of allowing for serial dependence in the observations is to relax the assumption that the parameter process is serially independent. A simple and mathematically convenient way to do so is to assume that it is a Markov chain. The chapter considers the likelihood of a hidden Markov model (HMM) in general. A very simple but crucial consequence of the matrix expression for the likelihood is the 'forward algorithm' for recursive computation of the likelihood. Such recursive computation plays a key role, not only in likelihood evaluation and hence parameter estimation, but also in forecasting, decoding and model checking.