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 ${\displaystyle X}$). An HMM requires that there be an observable process ${\displaystyle Y}$ whose outcomes depend on the outcomes of ${\displaystyle X}$ in a known way. Since ${\displaystyle X}$ cannot be observed directly, the goal is to learn about state of ${\displaystyle X}$ by observing ${\displaystyle Y}$. By definition of being a Markov model, an 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 ${\displaystyle t<t_{0}}$ must be conditionally independent of ${\displaystyle Y}$ at ${\displaystyle t=t_{0}}$ given ${\displaystyle X}$ at time ${\displaystyle t=t_{0}}$. 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.

Hidden Markov models are known for their applications to thermodynamics, statistical mechanics, physics, chemistry, economics, finance, signal processing, information theory, pattern recognition—such as speech recognition, handwriting recognition, gesture recognition, part-of-speech tagging, musical score following, partial discharges and bioinformatics.

Let ${\displaystyle X_{n}}$ and ${\displaystyle Y_{n}}$ be discrete-time stochastic processes and ${\displaystyle n\geq 1}$. The pair ${\displaystyle (X_{n},Y_{n})}$ is a hidden Markov model if

Let ${\displaystyle X_{t}}$ and ${\displaystyle Y_{t}}$ be continuous-time stochastic processes. The pair ${\displaystyle (X_{t},Y_{t})}$ is a hidden Markov model if

The states of the process ${\displaystyle X_{n}}$ (resp. ${\displaystyle X_{t})}$ are called hidden states, and ${\displaystyle \operatorname {\mathbf {P} } {\bigl (}Y_{n}\in A\mid X_{n}=x_{n}{\bigr )}}$ (resp. ${\displaystyle \operatorname {\mathbf {P} } {\bigl (}Y_{t}\in A\mid X_{t}\in B_{t}{\bigr )})}$ is called emission probability or output probability.

In its discrete form, a hidden Markov process can be visualized as a generalization of the urn problem with replacement (where each item from the urn is returned to the original urn before the next step).

In a room that is not visible to an observer there is a genie. The room contains urns X1, X2, X3, ... each of which contains a known mix of balls, with each ball having a unique label y1, y2, y3, ... . The genie chooses an urn in that room and randomly draws a ball from that urn. It then puts the ball onto a conveyor belt, where the observer can observe the sequence of the balls but not the sequence of urns from which they were drawn.

The genie has some procedure to choose urns: