Using pi symmetrization:
p{ij} = c{ij} / c_i where c_i = sumj c{ij}
\pi_j = \sum_j \pii p{ij}
x_{ij} = \pii p{ij} + \pij p{ji}
p^{rev}{ij} = x{ij} / x_i where x_i = sumj x{ij}
In words: takes the nonreversible transition matrix estimate, uses its
stationary distribution to compute an equilibrium correlation matrix,
symmetrizes that correlation matrix and then normalizes to the reversible
transition matrix estimate.
Using pi symmetrization: p{ij} = c{ij} / c_i where c_i = sumj c{ij} \pi_j = \sum_j \pii p{ij} x_{ij} = \pii p{ij} + \pij p{ji} p^{rev}{ij} = x{ij} / x_i where x_i = sumj x{ij}
In words: takes the nonreversible transition matrix estimate, uses its stationary distribution to compute an equilibrium correlation matrix, symmetrizes that correlation matrix and then normalizes to the reversible transition matrix estimate.