Multiplicatively closed Markov models must form Lie algebras
Group Theory
2017-09-04 v2 Statistics Theory
Populations and Evolution
Quantitative Methods
Statistics Theory
Abstract
We prove that the probability substitution matrices obtained from a continuous-time Markov chain form a multiplicatively closed set if and only if the rate matrices associated to the chain form a linear space spanning a Lie algebra. The key original contribution we make is to overcome an obstruction, due to the presence of inequalities that are unavoidable in the probabilistic application, that prevents free manipulation of terms in the Baker-Campbell-Haursdorff formula.
Cite
@article{arxiv.1704.01418,
title = {Multiplicatively closed Markov models must form Lie algebras},
author = {Jeremy G Sumner},
journal= {arXiv preprint arXiv:1704.01418},
year = {2017}
}
Comments
v2: 6 pages. Minimality condition included in Property 0 to close gap in the proof of main result. To appear in the ANZIAM Journal