Distributions associated with general runs and patterns in hidden Markov models

Abstract

This paper gives a method for computing distributions associated withpatterns in the state sequence of a hidden Markov model, conditionalon observing all or part of the observation sequence. Probabilitiesare computed for very general classes of patterns (competing patternsand generalized later patterns), and thus, the theory includes asspecial cases results for a large class of problems that have wideapplication.The unobserved state sequence is assumed to be Markovianwith a general order of dependence. An auxiliary Markov chain isassociated with the state sequence and is used to simplify the computations.Two examples are given to illustrate the use of the methodology.Whereas the first application is more to illustrate the basic stepsin applying the theory, the second is a more detailed applicationto DNA sequences, and shows that the methods can be adapted to includerestrictions related to biological knowledge.