The geometry of random drift V. Axiomatic derivation of the WFK diffusion from a variational principle

Abstract

An axiomatic derivation of the Wright–Fisher–Kimura (wfk) diffusion model for genetic drift is given using a variational principle. This is analogous to the characterization of the standard normal distribution in terms of an isoperimetric problem in the calculus of variations where the integrand is Shannon's information measure. We, on the other hand, give a Fisher-type information-theoretic interpretation of the variational principle largely motivated by the geometric approach to statistical likelihood theory due to S. V. Huzurbazzar, B. R. Rao and A. W. F. Edwards. In our process theory, the Ricci curvature tensor plays the role of information matrix. Ultimately, it is proved to be a normalized matrix of second partial derivates of the pseudodensity associated with the Christoffel velocity field. The proofs use classical projective differential geometry and depend on previous work in this series on the geometry of random drift.