| Hidden Markov models are a very rich class of models that have been used on problems as diverse as speech recognition, sodium ion channels, infectious disease processes, and rainfall occurrence. Since the late 1960's, however, it has been appreciated that even the seemingly mundane task of calculating the log-likelihood of hidden Markov models is not a trivial matter. This task, known as the evaluation problem, was addressed through the development of an early example of the Expectation-Maximization algorithm.;In this thesis, the problem of evaluation is addressed along both quantitative and qualitative lines. Quantitatively, an efficient algorithm is developed for fast computation of the log-likelihood, the score, and the observed information matrix from a single pass through the data. This enables one to readily obtain standard errors of parameter estimates, something that is rarely achieved in most typical EM settings. It also permits alternative maximization techniques to the EM algorithm. Qualitatively, sound goodness of fit techniques based on an expansion of the score are developed that are consistent against a wide variety of model mis-specifications. These techniques are shown to have good power in a range of situations, and may be performed with relatively little computational effort. A complementary goodness of fit method based on conditional residuals is also developed that enables one to effectively screen a wide variety of candidate predictor variables without requiring additional refitting of the model. The methods are evaluated through both simulations and real datasets. |
