HMMs

Hidden Markov Models (HMMs) are statistical models used to describe systems that are assumed to follow a Markov process with unobserved states. The model consists of hidden states that evolve over time and generate observable outputs. Each state has a probability distribution over possible observations, and the transition between states follows the Markov property, meaning the next state depends only on the current state.

An HMM is defined by a set of hidden states, an initial state distribution, a state transition probability matrix, and an emission distribution that links states to observations. Observations are produced at each time step, while the sequence of hidden states remains unseen. The model captures temporal structure and uncertainty in the data by combining state dynamics with probabilistic emissions.

Key tasks in working with HMMs include decoding, learning, and evaluation. Decoding seeks the most likely sequence of hidden states given the observed data, using algorithms such as Viterbi. Learning estimates the model parameters from data when the state sequence is unknown, typically via the Baum–Welch algorithm, an expectation–maximization method. Forward–backward procedures compute data likelihoods or posterior state probabilities.

HMMs accommodate discrete or continuous observations, with emission distributions ranging from multinomial to Gaussian or mixtures. Extensions include left-to-right (Bakis) models for speech, higher-order or factorial HMMs, and hidden semi-Markov models that model state durations. Limitations include reliance on the Markov and emission assumptions, potential training with local optima, and computational complexity for large models.

Applications span speech recognition, handwriting and gesture recognition, bioinformatics (e.g., gene finding), natural language processing (e.g., part-of-speech tagging), and time-series analysis in finance and engineering.