Learning Imprecise Hidden Markov Models
Arthur Van Camp, Erik Quaeghebeur, Filip Hermans · Ghent University Academic Bibliography (Ghent University) · 2011
Consider a stationary precise hidden Markov model (HMM) with n hidden states X k , taking values x k in a set {1, . . ., m} and n observations O k , taking values o k .Both the marginal model p X 1 (x 1 ), the emission models p O k |X k (o k |x k ) and the transition models p X k |X k-1 (x k |x k-1 ) are unknown.We can then use the Baum-Welch algorithm [see, e.g., 4] to get a maximum-likelihood estimate of these models.The Baum-Welch algorithm constructs the expected number of transitions) from state i to state j in the whole Markov chain.If we do not have enough data to justify a precise probability model, such as the one we obtain using the classical Baum-Welch algorithm, then the approach we present is useful.Our contribution exists of a method for learning imprecise transition probabilities in an HMM.We are not aware of another such method in the literature.These transitions from a state X k-1 = i to a state X k = j are multinomial processes.The imprecise Dirichlet model (IDM) is a convenient model for describing uncertainty about such processes [3].In order to learn using an IDM, we need the number of transitions and a choice for the pseudocounts s.Since the hidden states are unavailable, our method consists in taking the expected 1 number of transitions (positive real numbers instead of natural numbers), derived from the Baum-Welch algorithm, rather than real counts.So, the lower probability for state j conditional on state i is estimated by Q({ j}|i) := n i j/(s+∑ m