Hidden Markov Models

A hidden Markov model couples an unobserved Markov chain to observations whose distribution depends on the current hidden state, so the latent dynamics must be inferred indirectly from the data.

Definition

A hidden Markov model is a stochastic process in which an unobserved discrete-time Markov chain governs the regime, and at each time an observation is drawn from a distribution determined by the current hidden state, so the observations are conditionally independent given the hidden path.

Scope

This topic covers the doubly stochastic structure of a latent Markov chain with state-dependent emissions, the forward-backward algorithm for filtering and smoothing, the Viterbi algorithm for the most likely state path, parameter estimation by the Baum-Welch expectation-maximisation procedure, and the model's identifiability and asymptotic theory.

Core questions

• How does an unobserved Markov chain generate the observed sequence?
• How are the hidden state probabilities computed given the observations?
• How is the single most likely sequence of hidden states found?
• How are the model's transition and emission parameters estimated from data?

Key theories

Forward-backward algorithm

Recursive forward and backward passes compute the likelihood of the observations and the posterior distribution of each hidden state efficiently, enabling filtering and smoothing in time linear in the sequence length.

Baum-Welch parameter estimation

An expectation-maximisation procedure alternates between computing expected state occupancies and transitions under current parameters and re-estimating the transition and emission distributions, increasing the likelihood at each step.

Clinical relevance

Hidden Markov models are central to speech and handwriting recognition, computational biology including gene finding and sequence alignment, financial regime-switching models, and natural language processing, wherever an observed signal is driven by an unobserved sequence of discrete states.

History

The statistical foundations were laid by Baum and Petrie in the 1960s, the Viterbi algorithm for optimal decoding appeared in 1967 in a coding-theory context, and Rabiner's 1989 tutorial popularised the framework for speech recognition, after which hidden Markov models became standard across signal processing and bioinformatics.

Key figures

• Leonard Baum
• Ted Petrie
• Andrew Viterbi
• Lawrence Rabiner

Seminal works

• cappe2005
• rabiner1989

Frequently asked questions

What is hidden in a hidden Markov model?

The underlying Markov chain of states is not observed directly; only outputs whose distribution depends on the current state are seen, so the state sequence must be inferred from the observations.

What does the Viterbi algorithm compute?

It finds the single most probable sequence of hidden states given the observations, using dynamic programming to avoid enumerating exponentially many paths.