Hidden semi-Markov models (HSMMs)

A semi-Markov HMM (more properly called a hidden semi-Markov model, or HSMM) is like an HMM except each state can emit a sequence of observations. Let Y (Gt) be the subsequence emitted by “generalized state” Gt. The “generalized state” usually contains both the automaton state, Qt, and the length (duration) of the segment, Lt. We will define Y (Gt) to be the subsequence yt−l+1:t. After emitting a segment, the next state is Gtn , where tn = t + Lt. Similarly, denote the previous state by Gtp . Let Y (G + t ) be all observations following Gt, and Y (G − t ) be all observations preceeding Gt, as in Figure 1. Each segment Ot(q, l) def =P (Y (Gt)|Qt = q, Lt = l) can be an arbitrary distribution. If P (Y (Gt)|q, l) = ∏t i=t−l+1 P (yi|q), this is an explicit duration HMM [Fer80, Lev86, Rab89, MJ93, MHJ95]. If P (Y (Gt)|q, l) is modelled by an HMM or state-space model (linear-dynamical system), this is called a segment model [GY93, ODK96]. In computational biology, P (Y (Gt)|q, l) is often modelled by a weight matrix or higher-order Markov chain (see e.g., [BK97]). In this paper, we are agnostic about the form of P (Y (Gt)|q, l). It is possible to approximate a variable-duration HMM by adding extra states to a regular HMM (see [DEKM98, p69]), i.e., a mixture of geometric distributions. However, our main interest will be segment models, which are strict generalizations of variable-duration HMMs. For the relationship between semi-Markov HMMs, pseudo-2D HMMs, hierarchical HMMs, etc., please see [Mur02]. (Essentially, with a pseudo-2D HMMs, we know the size of each segment ahead of time; an HHMM is a generalization of a segment model where each segment can have subsegments inside of it, each modelled by an HMM.) We can represent a variable-duration HMMs as a DBN as shown in FIgure 2. We explicitly add Qt , the remaining duration of state Qt, to the state-space. (Even though Qt is constant for a long period, we copy its value across every