  We survey information-theoretic approaches to the reduction of Markov chains.
Our survey is structured in two parts: The first part considers Markov chain
coarse graining, which focuses on projecting the Markov chain to a process on a
smaller state space that is informative}about certain quantities of interest.
The second part considers Markov chain model reduction, which focuses on
replacing the original Markov model by a simplified one that yields similar
behavior as the original Markov model. We discuss the practical relevance of
both approaches in the field of knowledge discovery and data mining by
formulating problems of unsupervised machine learning as reduction problems of
Markov chains. Finally, we briefly discuss the concept of lumpability, the
phenomenon when a coarse graining yields a reduced Markov model.
