  We present a convex-concave reformulation of the reversible Markov chain
estimation problem and outline an efficient numerical scheme for the solution
of the resulting problem based on a primal-dual interior point method for
monotone variational inequalities. Extensions to situations in which
information about the stationary vector is available can also be solved via the
convex- concave reformulation. The method can be generalized and applied to the
discrete transition matrix reweighting analysis method to perform inference
from independent chains with specified couplings between the stationary
probabilities. The proposed approach offers a significant speed-up compared to
a fixed-point iteration for a number of relevant applications.
