Hidden Markov model

A hidden Markov model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process — call it ${\displaystyle X}$ — with unobservable ("hidden") states. As part of the definition, HMM requires that there be an observable process ${\displaystyle Y}$ whose outcomes are "influenced" by the outcomes of ${\displaystyle X}$ in a known way. Since ${\displaystyle X}$ cannot be observed directly, the goal is to learn about ${\displaystyle X}$ by observing ${\displaystyle Y.}$ HMM has an additional requirement that the outcome of ${\displaystyle Y}$ at time ${\displaystyle t=t_{0}}$ must be "influenced" exclusively by the outcome of ${\displaystyle X}$ at ${\displaystyle t=t_{0}}$ and that the outcomes of ${\displaystyle X}$ and ${\displaystyle Y}$ at ${\displaystyle t must be conditionally independent of ${\displaystyle Y}$ at ${\displaystyle t=t_{0}}$ given ${\displaystyle X}$ at time ${\displaystyle t=t_{0}.}$