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

Published online by Cambridge University Press:  01 July 2016

Peter L. Antonelli
Peter L. Antonelli*
Affiliation:
University of Alberta
*
∗Postal address: Department of Mathematics, The University of Alberta, Edmonton, Canada T6G 2G1.
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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.

Keywords

WFK DIFFUSION: WRIGHT–FISHER PROCESSCHRISTOFFEL VELOCITY FIELDRIEMANNIAN GEOMETRYPROJECTIVE DIFFERENTIAL GEOMETRYVARIATIONAL PRINCIPLEFISHER-TYPE INFORMATIONSTATIONARY DENSITY
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 3 , September 1979 , pp. 502 - 509
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

Research partially supported by NRC A-7667.

References

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