Hidden Markov Models
A Hidden Markov Model (HMM) is a probabilistic framework for modeling systems that evolve through sequences of states over time. The core principle is that the system operates through a series of hidden states that cannot be directly observed. Instead, an external observer only has access to outputs or observations generated from each state. This separation between hidden dynamics and observable outputs makes HMMs useful for problems where the true state of a system must be inferred from incomplete or noisy evidence.
Structure and Components
An HMM consists of several key components: a set of hidden states, a transition probability matrix that defines how the system moves between states, an emission probability matrix that defines what observations can be generated from each state, and an initial state distribution. The model assumes that future states depend only on the current state (the Markov property) and that observations depend only on the current state. These assumptions create a tractable framework for reasoning about sequential data.
Applications and Inference
HMMs are commonly used in speech recognition, biological sequence analysis, and financial forecasting. The primary inference problem in HMMs involves determining the most likely sequence of hidden states given a sequence of observations—a problem typically solved using the Viterbi algorithm. Forward and backward algorithms can also compute the probability of observations or estimate state probabilities at each time step. These computational methods make HMMs practical tools for real-world domains where understanding latent structure in sequential data is important.