Hidden Markov Model

A hidden Markov model consists of a hidden state sequence $\{Z_t\}$ following a Markov chain with transition matrix $A$ (entries $a_{ij}=P(Z_{t+1}=j\mid Z_t=i)$) and initial distribution $\pi$, together with an observed sequence $\{X_t\}$ where each $X_t$ is generated from an emission distribution depending only on the current hidden state, $P(X_t\mid Z_t=i)=b_i(X_t)$ (collected into an emission matrix $B$ for discrete observations, or emission densities for continuous ones). Crucially, $X_t$ is conditionally independent of everything else given $Z_t$. Three canonical computational problems define the standard HMM toolkit: evaluation (compute the likelihood of an observed sequence, via the forward algorithm), decoding (find the most likely hidden state sequence, via the Viterbi algorithm), and learning (estimate the parameters $A,B,\pi$ from observed data alone, via the Baum-Welch algorithm, a special case of the EM algorithm).

• The HMM is one of the most successful and widely deployed applied models in all of statistical machine learning, decades before deep learning, HMMs were the backbone technology for speech recognition systems, and remain the standard framework for gene and protein sequence annotation in bioinformatics.
• An HMM is a specific, discrete-hidden-state instance of the more general state-space model framework, the emission distribution corresponds to the observation equation and the hidden Markov chain to the state equation, the Kalman filter is the analogous continuous-state, linear-Gaussian special case.
• A common beginner confusion is between the forward-backward algorithm (used for smoothing, estimating the hidden state distribution at every time point given the full observed sequence) and the Viterbi algorithm (used for decoding, finding the single best hidden state path), they answer genuinely different questions and are not interchangeable.