🎲 Hidden Markov Model — States & Observations

In a regular Markov chain, states are directly observable. In an HMM, the states are hidden and only indirectly revealed through noisy or incomplete emissions. The model must infer the underlying state sequence from the observed output symbols.

The Viterbi algorithm efficiently finds the most probable sequence of hidden states given an observation sequence using dynamic programming. It runs in O(T·K²) time, where T is the sequence length and K is the number of states, making it tractable for practical problems.

HMMs are typically trained using the Baum–Welch algorithm, an instance of Expectation-Maximisation. It iterates between computing state occupation probabilities (E-step) and updating transition and emission parameters to maximise the likelihood of observed data (M-step).

HMMs formed the backbone of speech recognition from the 1980s through the 2000s. Modern systems use deep neural networks, but many hybrid architectures still combine neural acoustic models with HMM-based sequence decoding. Pure HMM systems remain in use for low-resource or embedded applications.

The Markov assumption states that the next state depends only on the current state, not the full history. It fails when long-range dependencies matter — for example, grammatical agreement over many words in language. Higher-order Markov models and neural sequence models relax this assumption.