Toric ideals of homogeneous phylogenetic models
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Error bounds for convolutional codes and an asymptotically optimum decoding algorithm
IEEE Transactions on Information Theory
Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals
Journal of Symbolic Computation
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A Viterbi path of length n of a discrete Markov chain is a sequence of n+1 states that has the greatest probability of occurring in the Markov chain. We divide the space of all Markov chains into Viterbi regions in which two Markov chains are in the same region if they have the same set of Viterbi paths. The Viterbi paths of regions of positive measure are called Viterbi sequences. Our main results are (1) each Viterbi sequence can be divided into a prefix, periodic interior, and suffix, and (2) as n increases to infinity (and the number of states remains fixed), the number of Viterbi regions remains bounded. The Viterbi regions correspond to the vertices of a Newton polytope of a polynomial whose terms are the probabilities of sequences of length n. We characterize Viterbi sequences and polytopes for two- and three-state Markov chains.