The Metropolis–Hastings algorithm transforms a given stochastic matrix into a reversible stochastic matrix with a prescribed stationary distribution. We show that this transformation gives the minimum distance solution in an $L^1$ metric.
"A Geometric Interpretation of the Metropolis-Hastings Algorithm." Statist. Sci. 16 (4) 335 - 339, November 2001. https://doi.org/10.1214/ss/1015346318