A Riemannian Optimization Approach for Finding the Nearest Reversible Markov Chain

We address the algorithmic problem of determining the reversible Markov chain (Formula presented.) that is closest to a given Markov chain (Formula presented.), with an identical stationary distribution. More specifically, (Formula presented.) is the reversible Markov chain with the closest transition matrix, in the Frobenius norm, to the transition matrix of (Formula presented.). To compute the transition matrix of (Formula presented.), we propose a novel approach based on Riemannian optimization. Our method introduces a modified multinomial manifold endowed with a prescribed stationary vector, while also satisfying the detailed balance conditions, all within the framework of the Fisher metric. We evaluate the performance of the proposed approach in comparison with an existing quadratic programming method and demonstrate its effectiveness through a series of synthetic experiments, as well as in the construction of a reversible Markov chain from transition count data obtained via direct estimation from a stochastic differential equation.