  We study dynamical reversibility in stationary stochastic processes from an
information theoretic perspective. Extending earlier work on the reversibility
of Markov chains, we focus on finitary processes with arbitrarily long
conditional correlations. In particular, we examine stationary processes
represented or generated by edge-emitting, finite-state hidden Markov models.
Surprisingly, we find pervasive temporal asymmetries in the statistics of such
stationary processes with the consequence that the computational resources
necessary to generate a process in the forward and reverse temporal directions
are generally not the same. In fact, an exhaustive survey indicates that most
stationary processes are irreversible. We study the ensuing relations between
model topology in different representations, the process's statistical
properties, and its reversibility in detail. A process's temporal asymmetry is
efficiently captured using two canonical unifilar representations of the
generating model, the forward-time and reverse-time epsilon-machines. We
analyze example irreversible processes whose epsilon-machine presentations
change size under time reversal, including one which has a finite number of
recurrent causal states in one direction, but an infinite number in the
opposite. From the forward-time and reverse-time epsilon-machines, we are able
to construct a symmetrized, but nonunifilar, generator of a process---the
bidirectional machine. Using the bidirectional machine, we show how to directly
calculate a process's fundamental information properties, many of which are
otherwise only poorly approximated via process samples. The tools we introduce
and the insights we offer provide a better understanding of the many facets of
reversibility and irreversibility in stochastic processes.
