Hidden Markov Models with Unobservable Transitions

We consider Hidden Markov Models (HMMs) that admit unobservable \(\varepsilon \) -transitions (also called null transitions), allowing state changes of which the observer is unaware. Due to the presence of \(\varepsilon \) -loops this additional feature complicates the theory and requires to carefully set up the corresponding probability space and random variables. In particular we present an algorithm for determining the most probable explanation given an observation (a generalization of the Viterbi algorithm for HMMs) and a method for parameter learning that adapts the probabilities of a given model based on an observation (a generalization of the Baum-Welch algorithm). The latter algorithm guarantees that the given observation has a higher (or equal) probability after adjustment of the parameters and its correctness can be derived directly from the so-called EM algorithm. We also provide runtime results in order to evaluate the efficiency of the proposed algorithms.