bibkey: “wolfer2021reversible” authors: “Geoffrey Wolfer; Shun Watanabe” year: 2021 title: “Information Geometry of Reversible Markov Chains” doi: “10.1007/s41884-021-00061-7” url: “https://doi.org/10.1007/s41884-021-00061-7” claim: “The reverse-relative-entropy projection onto reversible Markov kernels is the Perron stochastic rescaling of the entrywise geometric mean of a kernel and its time reversal, when their common support is strongly connected.” strata_touched: [] license: “citation-only” triage: “anchor”

Information Geometry of Reversible Markov Chains

Information Geometry 4, 393–433 (2021). Section 6, Theorem 7, on journal page 418 gives the e-projection and its Pythagorean identity; the proof continues on pages 419–420. The stochastic rescaling is defined in equation (3). For the e-projection, the intersection of the forward and reversed supports must be strongly connected. Strictly positive finite kernels satisfy this hypothesis. Time reversal is taken with respect to the kernel’s stationary law.

This is literature-attested background for the reversible Perron tilt in the parity-kernel testing proof. With uniform stationarity, the reversed kernel is the transpose and the geometric-mean matrix is symmetric. The projection construction is established literature; the rank-four factorization and the quartic depending on two parity-block averages are repo-derived specializations. The Bayes exponent proof separately controls its endpoint factor and supplies a subexponential lower bound.