Information Geometry of Reversible Markov Chains

Geoffrey Wolfer, Shun Watanabe

Research output: Contribution to journalArticlepeer-review


We analyze the information geometric structure of time reversibility for parametric families of irreducible transition kernels of Markov chains. We define and characterize reversible exponential families of Markov kernels, and show that irreducible and reversible Markov kernels form both a mixture family and, perhaps surprisingly, an exponential family in the set of all stochastic kernels. We propose a parametrization of the entire manifold of reversible kernels, and inspect reversible geodesics. We define information projections onto the reversible manifold, and derive closed-form expressions for the e-projection and m-projection, along with Pythagorean identities with respect to information divergence, leading to some new notion of reversiblization of Markov kernels. We show the family of edge measures pertaining to irreducible and reversible kernels also forms an exponential family among distributions over pairs. We further explore geometric properties of the reversible family, by comparing them with other remarkable families of stochastic matrices. Finally, we show that reversible kernels are, in a sense we define, the minimal exponential family generated by the m-family of symmetric kernels, and the smallest mixture family that comprises the e-family of memoryless kernels.

Original languageEnglish
Pages (from-to)393-433
Number of pages41
JournalInformation Geometry
Issue number2
StatePublished - 1 Dec 2021
Externally publishedYes


  • Exponential family
  • Information projection
  • Irreducible Markov chain
  • Mixture family
  • Reversible Markov chain

ASJC Scopus subject areas

  • Geometry and Topology
  • Statistics and Probability
  • Applied Mathematics
  • Computer Science Applications
  • Computational Theory and Mathematics


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