Hidden Markov Model
A Hidden Markov Model (HMM) is a statistical model of a system that transitions between unobserved (hidden) states over time, where each hidden state produces an observable output according to some probability distribution.
Definition
A Hidden Markov Model (HMM) is a statistical model of a system that transitions between unobserved (hidden) states over time, where each hidden state produces an observable output according to some probability distribution.
Overview
A Hidden Markov Model assumes a sequence of hidden states that evolve according to the Markov property — the next state depends only on the current state, not the full history — combined with a separate emission process that generates observable data from each hidden state. This two-layer structure (transition probabilities between hidden states, plus emission probabilities from hidden states to observations) makes HMMs useful whenever the true underlying process cannot be directly measured but leaves observable traces. Three canonical problems define how HMMs are used in practice. Evaluation asks how likely a given observation sequence is under a model, typically solved efficiently with the forward algorithm. Decoding asks what the most likely sequence of hidden states was, solved with the Viterbi algorithm, a dynamic-programming approach that avoids enumerating every possible state path. Learning asks how to estimate the model's transition and emission probabilities from data, usually via the Baum-Welch algorithm, a special case of expectation-maximization. HMMs were the workhorse of speech recognition for decades, used to model how hidden phonemes generate observed acoustic signals, and remain widely used in bioinformatics for gene finding and protein sequence alignment, as well as in finance for regime-switching models of asset returns. Their popularity in speech and language tasks declined with the rise of recurrent neural networks and later Transformer-based sequence models, which can capture longer-range dependencies without the Markov independence assumption. Still, HMMs remain valued for their interpretability, modest data requirements, and closed-form training guarantees, making them a common baseline and teaching tool before moving to deep sequence models.
Key Concepts
- Two-layer generative structure: hidden state transitions plus observable emissions
- Markov assumption — next hidden state depends only on the current state
- Forward algorithm for efficiently computing sequence likelihood
- Viterbi algorithm for finding the most probable hidden state sequence
- Baum-Welch (EM) algorithm for unsupervised parameter estimation
- Compact parameterization via transition and emission probability matrices
- Strong theoretical guarantees and interpretable state semantics
- Efficient dynamic-programming inference even for long sequences