Information geometry and population genetics. The mathematical structure of the Wright-Fisher model
type: Item
instance of: scholarly article
title: Information geometry and population genetics. The mathematical structure of the Wright-Fisher model
zbMATH Open document ID: 1370.92009
DOI: 10.1007/978-3-319-52045-2
author: Julian Hofrichter
author: Tat Dat Tran
author: Jürgen Jost
published in: Understanding Complex Systems
publication date: 2017-02-14T00:00:00Z
review text: The intention of the authors is to develop a new and systematic approach for the Wright-Fisher model, based on the geometric aspect as opposed to the stochastic or analytical structure of the model. This approach can unify many previous results and explore them from a general perspective, and therefore the authors do not delve too deeply into some of the different methods that have been applied to the Wright-Fisher models since its inception. A fundamental observation is that there exists a natural Riemannian metric on the probability simplex: \[ \overline{\Sigma}^n:=\{(p^0, p^1, \dots, p^n) : p^i \geq 0 \;\;\;\text{for all} \;i,\;\sum_{i=0}^n p^i=1 \}. \] This is the Fisher metric. It becomes singular on the boundary of the probability simplex \(\overline{\Sigma}^n\). More generally, the Fisher metric on a parametrized family of probability distributions measures how sensitively the family depends on the parameter when sampling from the underlying probability space. For multinomial distributions, the Fisher metrics is simply the inverse of the covariance matrix. This indicates that it is associated to iterate sampling from multinomial distributions. Therefore, the Kolmogorov equations can be interpreted as diffusion equations with respect to the Fisher metric. However, they do not model the Brownian motion for the Fisher metric. This gives a geometric approach to the exit times of the process, that is, the expected times of allele losses from the population. When considering exponential families, information geometry connects with the base quantities of statistical mechanics. These are entropy and free energy. The authors use the scheme and explore large deviation principles in the context of the Wright-Fisher model. The geometric structure behind the Kolmogorov equations also guides the analysis of the transitions between the different boundary strata of the simplex. The key is the degeneracy at the boundary of the Kolmogorov equations. For the Kolmogorov forward …
reviewed by: Fatima T. Adylova
Mathematics Subject Classification ID: 92-02
Mathematics Subject Classification ID: 92D10
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 94A17
zbMATH DE Number: 6684621
zbMATH Keywords: Wright-Fisher model
zbMATH Keywords: geometric structures
zbMATH Keywords: information geometry
zbMATH Keywords: multi-allelic models
zbMATH Keywords: Kolmogorov equations
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-52045-2
OpenAlex ID: W2589557898
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description: scientific article; zbMATH DE number 6684621