On the definitions of hidden Markov models

Abstract

At least three probabilistic definitions of hidden Markov models (HMMs) have been used frequently in the literature. Unfortunately, one of these definitions shows fatal flaws, however nowadays a lot of literature still uses this definition. The aim of this paper is on one hand to specifically point out one such fatal flaw (in terms of deriving the well-known forward-backward algorithm), and on the other hand to list key properties of HMMs under the other two plausible (and equivalent) probabilistic definitions for further developments. As applications, we rigorously layout forward-backward algorithms for inhomogeneous HMMs and hidden reciprocal models, and fully present connections between HMMs and undirected graphical models which are not mentioned anywhere in the literature, to the best of our knowledge.