Introduction Programming hidden Markov models

Since their formulation by Andrei Markov in 1906 [7], Markov chains (MC) and hidden Markov models (HMM) have found a place in diverse fields of science and engineering, from speech recognition to weather prediction to protein sequence alignment. Wherever a data set can be expressed as a string of discrete symbols, and when the data has a common source or common underlying principle, then for those cases a hidden Markov model may be designed to extract that underlying principle. A Markov chain is a directed graph where the nodes are Markov states and the edges are directed state-state transitions. A Markov state is said to ‘emit’ a symbol, which is unique to that state. A “hidden” Markov model (also sometimes called a “state space” or “latent Markov” model) differs from a MC in that each state emits one of a set of symbols drawn from a distribution; different hidden states may emit the same symbol, and non-emitting states are possible. The term ‘hidden’ is used because the symbol sequence alone does not tell us the state sequence directly. Instead, the latter must be inferred. In a HMM the states have a meaning all their own, separate from the meaning of the symbols they emit. For example, in a HMM composed of states that emit temperature readings, the states themselves may represent precipitation readings, or wind direction, or seasons, or all of the above. HMM states are classifiers of the symbols in the data string(s), their types and their contexts. Algorithms for computing the probabilistic fit between a data string, or a set of strings, and a HMM have long-since been worked out. The groundbreaking work of L.E. Baum in the 1960’s led to the expectation-maximization (EM) method for locally optimizing HMM parameters. In 1967, Andrew Viterbi wrote a general algorithm for finding the optimal state pathway given a sequence [8]. Lawrence Rabiner’s highly-cited 1989 tutorial [6] outlined the “Three Basic Problems” for HMMs (see box), and brought these techniques within reach of scientists not traditionally trained in probability theory, including even biologists (who then became known as “bioinformaticists”). Several good books on the subject are now available [1–5]. But the algorithmology of HMMs still has many unsolved problems, some of which are addressed in the current special issue. For example, the space of all possible directed graphs of size Q may be very, very large, far too large to be searched exhaustively. How do we find the optimal graph connectivity in a datadriven manner? How do we, simultaneously, define the number of states and initialize their emission probabilities, also in a data driven manner? One theoretical approach is presented in this issue (see the article by Li and Biswas). A related problem is to determine the Markov chain order, given only the data. In a firstorder chain, the transition probability depends only on the current state; in a second order model it depends also on the previous two states, and so on. An answer is given in this issue (see the article by Boys and Henderson). Also, how do we impose constraints from domain-specific knowledge on the HMM topology? In this issue we present the special cases of psychological data and speech recognition (see articles by Visser et al., and by Abdulla, respectively). The Three Basic Problems for HMMs [6]