  Symbolic dynamics has proven to be an invaluable tool in analyzing the
mechanisms that lead to unpredictability and random behavior in nonlinear
dynamical systems. Surprisingly, a discrete partition of continuous state space
can produce a coarse-grained description of the behavior that accurately
describes the invariant properties of an underlying chaotic attractor. In
particular, measures of the rate of information production--the topological and
metric entropy rates--can be estimated from the outputs of Markov or generating
partitions. Here we develop Bayesian inference for k-th order Markov chains as
a method to finding generating partitions and estimating entropy rates from
finite samples of discretized data produced by coarse-grained dynamical
systems.
