Hidden Markov model

A hidden Markov model (HMM) is a Markov model in which the observations are dependent on a latent (or hidden) Markov process (referred to as X {\displaystyle X} ). An HMM requires that there be an observable process Y {\displaystyle Y} whose outcomes depend on the outcomes of X {\displaystyle X} in a known way. Since X {\displaystyle X} cannot be observed directly, the goal is to learn about state of X {\displaystyle X} by observing Y {\displaystyle Y} . By definition of being a Markov model, an HMM has an additional requirement that the outcome of Y {\displaystyle Y} at time t = t 0 {\displaystyle t=t_{0}} must be "influenced" exclusively by the outcome of X {\displaystyle X} at t = t 0 {\displaystyle t=t_{0}} and that the outcomes of X {\displaystyle X} and Y {\displaystyle Y} at t < t 0 {\displaystyle t<t_{0}} must be conditionally independent of Y {\displaystyle Y} at t = t 0 {\displaystyle t=t_{0}} given X {\displaystyle X} at time t = t 0 {\displaystyle t=t_{0}} . Estimation of the parameters in an HMM can be performed using maximum likelihood estimation. For linear chain HMMs, the Baum–Welch algorithm can be used to estimate parameters.