Hidden Markov Model

In simple terms

Some sequences are produced by a process you can’t observe directly — only its effects. A person speaking is in a sequence of phoneme states; you hear sound waves, not the phonemes. An HMM models this: there is a hidden chain of states (the phonemes, or a market regime, or a gene), each state emits observations with some probability, and the states transition to each other with known probabilities. Given the observations, you can infer what the hidden states were.

More detail

An HMM is defined by:

• States S = {s₁, …, sN} — the hidden process (finite state machine).
• Transition matrix A[i][j] = P(next state = j | current state = i) — each row sums to 1.
• Emission matrix B[i][o] = P(observing o | current state = i) — for discrete observations.
• Initial distribution π[i] = P(start in state i).

Three canonical algorithms:

• Forward algorithm — computes P(observations | model) efficiently using dynamic programming. Used for likelihood evaluation.
• Viterbi algorithm — finds the most likely hidden state sequence given observations. Used for decoding (e.g., which phonemes produced this audio?).
• Baum-Welch algorithm — learns model parameters (A, B, π) from unlabelled observation sequences using Expectation-Maximisation. The E-step runs Forward-Backward; the M-step re-estimates parameters.

HMMs extend to continuous observations (Gaussian emission, used in speech recognition) and hierarchical structures. They were the dominant model for speech recognition from the 1970s through ~2015, when deep neural networks (CTC, sequence-to-sequence models) took over.

The Markov property underlying HMMs — the next state depends only on the current state, not the full history — is both the source of their tractability and their limitation. Real sequences often have long-range dependencies that an HMM cannot capture without an exponential number of states.

Why it matters

HMMs introduced dynamic programming as a core tool for sequence modelling, and the three algorithms (Forward, Viterbi, Baum-Welch) appear in many guises throughout ML: Conditional Random Fields, Kalman filters, and belief propagation are all generalisations. Understanding HMMs makes the attention mechanism in transformers more legible — both solve the problem of relating each position in a sequence to all others, just in very different ways.

Real-world examples

• Siri and Google Voice used HMM-based acoustic models for speech recognition through the early 2010s.
• Bioinformatics uses profile-HMMs to detect remote homologues of protein families (HMMER).
• Part-of-speech tagging and named-entity recognition used HMMs before neural sequence models became standard.
• Financial regime detection (bull/bear market hidden states from price observations).

Common misconceptions

• “HMMs are obsolete.” In speech recognition, yes — neural models dominate. In bioinformatics and some NLP tasks, HMMs remain competitive and interpretable.
• “The Markov assumption means the model has no memory.” It means the next state depends on the current state only, but the current state can encode whatever history was relevant to reach it.

Learn next

HMMs are a probabilistic sequence model; transformers are the successor that handles long-range dependencies far better. The probabilistic framing comes from probability and statistics; the sequence labelling use case leads to natural language processing.