bibkey: dyergreenhillullrich2014heatbath
authors: Martin Dyer; Catherine Greenhill; Mario Ullrich
year: 2014
title: Structure and eigenvalues of heat-bath Markov chains
doi: 10.1016/j.laa.2014.04.018
url: https://arxiv.org/abs/1301.4055v3
claim: A state-independent mixture of heat-bath conditional-resampling kernels has nonnegative transition spectrum; this does not compare its two nontrivial spectral edges.
strata_touched: []
license: citation-only
triage: anchor

Nonnegative spectrum of heat-bath mixtures

The primary source is Dyer–Greenhill–Ullrich, Linear Algebra and its Applications 454, 57–71, DOI:10.1016/j.laa.2014.04.018, arXiv:1301.4055v3, Definition 1.1 and Lemma 1.2. This note does not redistribute the source text or PDF.

Each heat-bath update resamples within a conditional class from the stationary law. It is idempotent and reversible. The source proves that a state-independent mixture of these updates has no negative eigenvalues.

For a finite positive legal-word law, the FIB site update resamples each legal toggle pair and keeps blocked singleton words fixed. With $D=\mathrm{diag}(\sqrt{p})$ and the symmetric site-parent operator $H_{\omega }$, its transition matrix is $T_{\omega }=I-D^{-1}{H}_{\omega }D$. The cited result gives $0⪯H_{\omega }⪯I$ after symmetrization. It does not give the stronger comparison $\gamma (H_{\omega })+\Lambda (H_{\omega })\le 1$, nor the associated Fisher-design optimization.

The transition operator is a classical conditional update. Its diagonal symmetrization does not identify that stochastic update with real-time unitary motion.