Hidden Markov Models

A hidden Markov model (HMM) is a probabilistic model of sequences in which a process with hidden states—not directly observed—generates a sequence of visible observations. At each step, the state changes according to transition probabilities and produces an observation from a distribution associated with that state; the task is to reason about the hidden process from what can be observed.

What defines it

An HMM is specified by four elements: the set of hidden states; the transition probabilities between states; the emission distributions, which connect each state to possible observations; and the initial distribution. The first-order Markov assumption says that, given the current state, the probability of the next state does not need the entire state history.

A primary example shows how the abstraction becomes a model. In an AT&T Bell Laboratories speech recognizer, Levinson, Ljolje, and Miller represented unobservable phonetic units as states, acoustic measurements as observations, and temporal structure through transitions and durations. A hidden state is not a label disclosed by the data: it is a modeling hypothesis whose usefulness depends on whether emissions and transitions describe the sequence well.

The three classic problems

Using an HMM usually separates three questions. Evaluation sums the probability of an observed sequence over compatible hidden paths, commonly through the forward recursion. Decoding seeks a maximum-probability state path; the speech system above used a modified Viterbi algorithm and retained decisions to reconstruct the best sequence. Evaluation and decoding are not the same operation: one sums paths, while the other selects one.

Learning fits transition and emission parameters to observations. The name Baum–Welch refers to a family of forward–backward re-estimation procedures: Baum, Petrie, Soules, and Weiss published an iterative maximization technique in 1970 for probabilistic functions of Markov chains. Iteration improves the objective from its starting point, but that does not automatically make the result the only explanation of the data: different states can model similar patterns.

Uses and current status

HMMs have verifiable sequence applications. In 1989, Huang, Hon, and Lee documented a large-vocabulary continuous-speech recognizer based on semi-continuous HMMs. In 2016, Stratos, Collins, and Hsu trained anchor HMMs for unsupervised part-of-speech tagging and noted that their states could be lexicalized by words, a concrete form of interpretability.

Neural networks opened another route, not a universal expiry date. Graves and Jaitly presented an end-to-end recurrent system in 2014 that transcribed audio without an intermediate phonetic representation. That alternative reduces hand-designed components, whereas an HMM retains an explicit separation between state, transition, and emission. The choice depends on the data, the task, and whether that discrete structure supplies a useful explanation.