  The problem of calculating the rate of mutual information between two
coarse-grained variables that together specify a continuous time Markov process
is addressed.
  As a main obstacle, the coarse-grained variables are in general
non-Markovian, therefore, an expression for their Shannon entropy rates in
terms of the stationary probability distribution is not known. A numerical
method to estimate the Shannon entropy rate of continuous time hidden-Markov
processes from a single time series is developed.
  With this method the rate of mutual information can be determined
numerically. Moreover, an analytical upper bound on the rate of mutual
information is calculated for a class of Markov processes for which the
transition rates have a bipartite character. Our general results are
illustrated with explicit calculations for four-state networks.
