Hidden Markov chain

A hidden Markov chain $Z$ (or a function of a Markov chain) is a stochastic process of the form $Z=\Phi(Y)$, where $Y$ is a Markov chain over alphabet $\mathcal{Y}$ and $\Phi$ is a deterministic function defined on $\mathcal{Y}$ with values in $\mathcal{Z}$.

Alternatively, a hidden Markov chain is defined as the output process when passing a Markov chain through a noisy channel. It is well known that the two definitions are equivalent if $\mathcal{Y}$ is finite.