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Baum-Welch Algorithm
An expectation-maximization procedure for estimating a hidden Markov model's transition and emission parameters purely from observed data, without ever seeing the hidden states directly.
In Plain English
If you know a hidden Markov model's transition and emission rules, computing things about the hidden states is a solved problem. But usually you don't know those rules, you only have a stream of observations and want to learn them. The Baum-Welch algorithm solves this chicken-and-egg problem iteratively: given a current guess at the parameters, use the forward-backward algorithm to estimate how likely each hidden state was at each time point (as if you knew the parameters), then use those soft estimates to refine the parameter guess (as if you knew the hidden states), and repeat until the estimates stop changing. Each round is guaranteed to make the observed data at least as likely as before, never less.
Definition
The Baum-Welch algorithm is a special case of the Expectation-Maximization (EM) algorithm applied to hidden Markov models, used to obtain maximum likelihood estimates of the transition matrix , emission parameters , and initial distribution from an observed sequence alone. In the E-step, given current parameter estimates, the forward-backward algorithm computes the posterior probabilities of each hidden state (and each state-transition) at every time point given the entire observed sequence. In the M-step, these posterior probabilities are used as soft (probabilistic) counts to re-estimate , , and via their maximum-likelihood formulas, exactly as if the hidden states had actually been observed, but weighted by how likely each hidden state assignment is. Like the general EM algorithm, Baum-Welch is guaranteed to monotonically increase the observed-data log-likelihood at every iteration and converges to a local (not necessarily global) maximum, making the choice of starting parameters and multiple random restarts a standard practical consideration.
Properties
- Baum-Welch is unsupervised parameter learning in the truest sense: it needs no labeled examples of which hidden state produced which observation, only the raw observed sequence and a chosen number of hidden states, everything else is inferred purely from the data's internal statistical structure.
- Because Baum-Welch only guarantees convergence to a local likelihood maximum (a general EM property, not specific to HMMs), the final parameter estimates can depend meaningfully on the starting values, running the algorithm from several different random initializations and keeping the best result is standard, recommended practice.
- Named after Leonard Baum and Lloyd Welch, who developed the technique in the late 1960s at the Institute for Defense Analyses, years before the general EM algorithm was formally named and proven by Dempster, Laird, and Rubin in their landmark 1977 paper, the HMM-specific case actually predates the general framework it's now understood as an instance of.
At a Glance