Papers › A Riemannian Optimization Approach for Finding the Nearest Reversible Markov Chain
A Riemannian Optimization Approach for Finding the Nearest Reversible Markov Chain
Fabio Durastante, Miryam Gnazzo, Beatrice Meini
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We address the algorithmic problem of determining the reversible Markov chain X̃ that is closest to a given Markov chain X, with an identical stationary distribution. More specifically, X̃ is the reversible Markov chain with the closest transition matrix, in the Frobenius norm, to the transition matrix of X. To compute the transition matrix of X̃, 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.
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