Rabiner 1989 - HMM Tutorial - Overview

Summary

Rabiner (1989), A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition (Proc. IEEE 77(2):257-286), is the standard exposition of the HMM and its three problems: evaluation (forward-backward), decoding (posterior argmax and Viterbi) and learning (Baum-Welch EM). It also covers model types (ergodic, left-right), continuous densities, explicit state duration, scaling, multiple sequences, initialisation, and speech-recognition applications. For Bayesian work, it supplies the exact likelihood recursion that PyMC/Stan marginalise over, and the Viterbi and posterior decoders applied per posterior draw.

Overview

Research question. How can a signal be modelled as a probabilistic function of an unobserved Markov chain, and how can the three practical problems (likelihood, best state sequence, parameter fitting) be solved efficiently?

Contributions. A self-contained derivation of discrete Markov chains and HMMs, the forward-backward procedure ($O(N^{2}T)$ vs $O(T{N}^T)$ enumeration), the Viterbi algorithm, Baum-Welch reestimation as EM, practical issues (scaling, multiple observation sequences, initial estimates, insufficient training data, model choice), extensions (continuous mixtures, autoregressive HMMs, null transitions, parameter tying, explicit durations, ML/MMI/MDI criteria) and three speech applications with error-rate tables.

Key result for this vault. Evaluation by forward recursion cost: with $N=5$, $T=100$ the direct sum needs about $10^{72}$ operations while the forward procedure needs about 3000, “a savings of about 69 orders of magnitude”.

Source Map

Paper section	Content	Note
I Introduction	signal models, three problems, organisation	this note, Rabiner 1989 - Markov Chains, HMM Definition and the Three Problems
II Discrete Markov processes (eqs 1-6, Figs 1-3)	chain, weather example, duration density, coin and urn models	Rabiner 1989 - Markov Chains, HMM Definition and the Three Problems
II.A-II.C Elements of an HMM, three problems (eqs 7-11)	$\lambda =(A,B,\pi )$	same
III.A Solution to Problem 1 (eqs 12-25, Figs 4-5)	forward, backward	Rabiner 1989 - Problem 1 - Forward and Backward Algorithms
III.B Solution to Problem 2 (eqs 26-35)	$\gamma$, Viterbi	Rabiner 1989 - Problem 2 - State Posteriors and the Viterbi Algorithm
III.C Solution to Problem 3 (eqs 36-44, Fig. 6)	$\xi$, Baum-Welch, EM	Rabiner 1989 - Problem 3 - Baum-Welch Reestimation
IV Types of HMMs (eqs 45-83, Figs 7-9)	ergodic, left-right, mixtures, AR, null, tying, duration	Rabiner 1989 - HMM Types, Continuous Densities and State Duration
IV.E-F Optimisation criteria, comparison (eqs 84-89)	ML, MMI, MDI, distance	same
V Implementation issues (eqs 90-113, Fig. 10)	scaling, multiple sequences, initial estimates, insufficient data, model choice	Rabiner 1989 - Implementation Issues - Scaling, Multiple Sequences and Initialisation
VI Speech recognisers (eqs 114-119, Figs 11-20, Table 1)	isolated word, LPC, VQ, k-means	Rabiner 1989 - Speech Recognition Applications and Results
VII Connected words (eqs 120-125, Figs 21-23, Table 2)	level building	same
VIII Large vocabulary, limitations, IX Summary, references		same

Motivating PyMC examples

From the examples page:

• E24 Hidden Markov models (USGS M7+ earthquake counts 1900-2024; elk GPS tracks): forward algorithm in pytensor.scan vs pymc_extras DiscreteMarkovChain + marginalize, label switching and ordering, forward-backward and Viterbi per draw, dwell times, pseudo-residuals, integrated LOO.
• E48 Bayesian NHMM for Lorenz-84 regimes (nonhomogeneous HMM with seasonal softmax transitions).
• E59 kinesin stepping (hypoexponential dwells, 2-phase lattice HMM forward algorithm).
• E57 single-particle tracking (diffusion-state mixtures, GPB1 switching HMM). Bayesian extensions of this source are covered in Bayesian HMMs - Label Switching, Non-Homogeneous Transitions and Dwell Times (concept note, no raw source).

Connections

Existing vault: Model Building with Latent Variables - Animal Movement (movement HMMs), The Kalman Filter and Marginal Likelihood via the Kalman Filter (the continuous-state analogue of the forward recursion), State-Space Models and the Kalman Filter - Overview. Discrete latents and marginalisation: Marginalising Discrete Latents - Why NUTS Cannot Sample Discrete Parameters.

Relevance to marketing measurement

HMMs model customer or market regimes: engagement states with Markov transitions, promotion regimes, dormant/active customers, and time-varying channel effectiveness with discrete regimes. The forward algorithm gives exact marginal likelihoods for gradient-based Bayesian fitting; Viterbi/posterior decoding labels the regime history; explicit duration (Sec. IV.D) matters for campaigns whose regimes are not geometric. Non-homogeneous extensions let covariates (seasonality, spend) drive transitions.

See Also

Model Building with Latent Variables - Animal Movement, The Kalman Filter, Marginalising Discrete Latents - Why NUTS Cannot Sample Discrete Parameters