# Multiplicatively closed Markov models must form Lie algebras

@article{Sumner2017MultiplicativelyCM, title={Multiplicatively closed Markov models must form Lie algebras}, author={Jeremy G. Sumner}, journal={Anziam Journal}, year={2017}, volume={59}, pages={240-246} }

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 with 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, which prevents free manipulation of terms in the Baker–Campbell–Haursdorff formula.

## 11 Citations

### Uniformization stable Markov models and their Jordan algebraic structure

- Mathematics
- 2021

This work provides a characterisation of the continuous-time Markov models where there exists a elementary relationship between the associated Markov (stochastic) matrices and the rate matrices (generators) and shows that the existence of an underlying Jordan algebra provides a sufficient condition, which becomes necessary for (socalled) linear models.

### Lie-Markov Models Derived from Finite Semigroups

- MathematicsBulletin of mathematical biology
- 2019

A general method for deriving a Lie-Markov model from a finite semigroup that is a continuous-time Markov chain on k-states and satisfies the property of multiplicative closure.

### 2 S ep 2 01 7 Lie-Markov models derived from finite semigroups

- Mathematics
- 2018

We present and explore a general method for deriving a Lie-Markov model from a finite semigroup. If the degree of the semigroup is k, the resulting model is a continuous-time Markov chain on k states…

### Closed codon models: just a hopeless dream?

- Mathematics
- 2018

Investigation of toy models indicated that finding the Lie closure of matrix linear spaces which deviated only slightly from a simple model resulted in a Lie closure that was close to having the maximum number of parameters possible, and further consideration of the variants of linearly closed models is proposed.

### Phylosymmetric Algebras: Mathematical Properties of a New Tool in Phylogenetics

- MathematicsBulletin of mathematical biology
- 2020

This work explores a method of building a rate matrix set from a rooted tree structure by assigning rates to internal tree nodes and states to the leaves, then defining the rate of change between two states as the rate assigned to the most recent common ancestor of those two states.

### The impracticalities of multiplicatively-closed codon models: a retreat to linear alternatives

- MathematicsJournal of Mathematical Biology
- 2020

It was hypothesised that finding the Lie closure of a codon model could help to solve the problem of mis-estimation of the non-synonymous/synonymous rate ratio, and two different methods of finding a linear space from a model are proposed.

### Systematics and symmetry in molecular phylogenetic modelling: perspectives from physics

- BiologyJournal of Physics A: Mathematical and Theoretical
- 2019

The aim of this review is to present and analyze the probabilistic models of mathematical phylogenetics, as the foundations on which the practical implementations are based, and to emphasize the many features of multipartite entanglement which are shared between descriptions of quantum states on the physics side, and the multi-way tensor probability arrays arising in phylogenetics.

### Exploring the consequences of lack of closure in codon models

- Biology
- 2017

Simulation is used to investigate the accuracy of estimation of both the selection parameter $omega and branch lengths in cases where the underlying DNA process is heterogeneous but $\omega$ is constant, and it is found that both $\omegas$ and branch length can be mis-estimated in these scenarios.

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