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| Baum–Welch algorithm | |
|---|---|
| Name | Baum–Welch algorithm |
| Developer | L. E. Baum, Lloyd R. Welch |
| Introduced | 1970s |
| Paradigm | Expectation–maximization algorithm, Hidden Markov model, stochastic process |
| Input | Observed sequence, model order |
| Output | Maximum likelihood parameters for a Hidden Markov model |
Baum–Welch algorithm The Baum–Welch algorithm is an iterative procedure for estimating parameters of a Hidden Markov model from observed data using a variant of the Expectation–maximization algorithm. It was developed by L. E. Baum and Lloyd R. Welch and has been widely applied in fields ranging from speech recognition and bioinformatics to natural language processing and signal processing. The method alternates between computing expected sufficient statistics under the current model and maximizing likelihood estimates, producing locally optimal parameter sets.
The algorithm addresses parameter estimation for a model defined by a set of hidden states, transition probabilities, and emission probabilities within a Hidden Markov model framework. It operates by employing forward–backward recursions to compute state occupancy and transition expectations for observed sequences as in early work by Rabiner, Baum, and contemporaries, then updates parameters to improve likelihood. Practical deployments include systems developed by organizations like Bell Labs, techniques in genomics research from groups at Broad Institute and European Bioinformatics Institute, and pipelines in industrial projects by Google, Microsoft Research, and IBM Research.
Consider a discrete-time Hidden Markov model with N hidden states and M observation symbols. Let the transition matrix be A = {a_ij}, the emission matrix be B = {b_j(k)}, and the initial state distribution be pi = {pi_i}. Given an observation sequence O = (O_1,...,O_T), the goal is to maximize the likelihood P(O | A, B, pi). The forward variable alpha_t(i) = P(O_1,...,O_t, q_t = s_i | model) and the backward variable beta_t(i) = P(O_{t+1},...,O_T | q_t = s_i, model) are computed by dynamic programming analogous to recursions used in work at Bell Labs and detailed in tutorials by Rabiner and others. Expected state occupation gamma_t(i) and expected state-to-state transitions xi_t(i,j) are defined using alpha and beta and are used to form re-estimation formulas for a_ij, b_j(k), and pi_i, paralleling methods compared in surveys from IEEE and texts from MIT Press and Springer.
The Baum–Welch procedure is an instantiation of the Expectation–maximization algorithm specialized to Hidden Markov model latent structures. In the E-step, compute Q(θ, θ_old) = E_{Z|O,θ_old}[log P(O,Z | θ)] by evaluating expected counts gamma_t(i) and xi_t(i,j) via forward–backward probabilities. In the M-step, maximize Q with respect to θ to obtain re-estimated transition and emission probabilities that increase P(O | θ). This derivation mirrors the general EM treatments in monographs from Dempster–Laird–Rubin, lectures at Stanford University and MIT, and expositions in journals such as Journal of the Royal Statistical Society and IEEE Transactions on Information Theory.
Implementation requires careful computation of alpha and beta arrays across time steps 1..T and states 1..N. Pseudocode follows a loop: initialize A, B, pi (randomly or via heuristics from corpora at Linguistic Data Consortium), repeat until convergence: compute alpha via forward recursion, compute beta via backward recursion, compute gamma and xi, re-estimate pi = gamma_1, a_ij = sum_t xi_t(i,j)/sum_t gamma_t(i), b_j(k) = sum_{t: O_t=k} gamma_t(j)/sum_t gamma_t(j). Practical code appears in toolkits from HTK and Kaldi and in libraries maintained by NumPy and SciPy-based projects. Pseudocode is analogous to treatments in textbooks by Christopher M. Bishop, Dan Jurafsky, and James P. Allison.
Baum–Welch guarantees non-decreasing likelihood but can converge to local maxima, a behavior noted in comparisons by researchers at Bell Labs, AT&T Laboratories, and Carnegie Mellon University. Numerical underflow in alpha and beta recursions is mitigated via scaling factors or log-space implementations as recommended in work from Rabiner and implemented in frameworks by Google DeepMind and OpenAI tooling. Regularization and prior distributions inspired by methods from Bayesian statistics groups at Oxford University and University of Cambridge reduce overfitting; model selection can use criteria like Akaike Information Criterion and Bayesian Information Criterion evaluated in studies published in IEEE Transactions on Pattern Analysis and Machine Intelligence.
Classic applications include phoneme recognition in systems developed at Bell Labs and Xerox PARC, gene prediction and sequence alignment in projects at National Institutes of Health and European Molecular Biology Laboratory, part-of-speech tagging in corpora from Penn Treebank and Brown Corpus, and financial time-series analysis used by banks like Goldman Sachs and JPMorgan Chase. Example use-cases in bioinformatics include hidden-state models for protein secondary structure prediction explored at Sanger Institute and motif discovery published by groups at Stanford University and EMBL-EBI.
Extensions include continuous-observation HMMs with Gaussian mixtures as in systems by HTK and Kaldi, discriminative training algorithms like Maximum Mutual Information and Conditional Random Field approaches compared in studies at Microsoft Research and Google Research, Bayesian nonparametric variants such as the Hierarchical Dirichlet Process HMM developed in work from University of California, Berkeley and University of Toronto, and stochastic EM or variational EM adaptations used in large-scale settings at Amazon and Facebook AI Research. Connections exist to algorithms for state estimation like the Viterbi algorithm and to broader latent-variable methods treated in monographs from Springer and Cambridge University Press.
Category:Algorithms