Information geometry and population genetics. The mathematical structure of the Wright-Fisher model
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 equation, it is gained a global solution concept from the equations for the moments of the process, which incorporates the dynamics on the entire simplex, including all boundary strata. This also involves the duality between the Kolmogorov forward and backward equations. The stationary case is described by an elliptic equation whose solutions arise from the time-dependent equation as time goes to infinity. From the information geometric perspective, one can interpret these solutions naturally in terms of entropies. Thus, the authors mainly focus on the basic Wright-Fisher model in the absence of selection or mutation. The moment generating function and energy functional method are systematically applied.
scientific article; zbMATH DE number 6684621
scientific article