  We analyze the information geometric structure of time reversibility for
parametric families of irreducible transition kernels of Markov chains. We
define and characterize reversible exponential families of Markov kernels, and
show that irreducible and reversible Markov kernels form both a mixture family
and, perhaps surprisingly, an exponential family in the set of all stochastic
kernels. We propose a parametrization of the entire manifold of reversible
kernels, and inspect reversible geodesics. We define information projections
onto the reversible manifold, and derive closed-form expressions for the
e-projection and m-projection, along with Pythagorean identities with respect
to information divergence, leading to some new notion of reversiblization of
Markov kernels. We show the family of edge measures pertaining to irreducible
and reversible kernels also forms an exponential family among distributions
over pairs. We further explore geometric properties of the reversible family,
by comparing them with other remarkable families of stochastic matrices.
Finally, we show that reversible kernels are, in a sense we define, the minimal
exponential family generated by the m-family of symmetric kernels, and the
smallest mixture family that comprises the e-family of memoryless kernels.
