Branching Processes
10 Apr 2009 17:40
A class of stochastic process important as models in genetics and population biology, chemical kinetics, and filtering. The basic idea is that there are a number of objects, often called particles, which, in some random fashion, reproduce ("branch") and die out; they can be of multiple types and occupy differing spatial locations. They can pursue their trajectories and their biographies either independently, or with some kind of statistical dependence across particles.
The most basic version has one type of particle, and no spatial considerations. At each time step, each parrticle gives rise to a random number of offspring; the distribution of offspring is fixed, and the number is independent across time-steps and across lineages (IID). This is the so-called Galton-Watson branching process. Galton introduced it as a model of the survival of (patrilneal) family names, so that only male offspring counted; he required the distribution of time until a given lineage went extinct. This was provided almost immediately by Watson, in a very elegant use of the method of generating functions, which is, itself, reproduced in probability textbooks down to the present day. (However, when I first encoutnered the problem, in a probability class, the teacher presented it as one about the survival of matrilineal lineages, defined by inheritance of mitochondrial DNA. Whether this was conscious subversion of the patriarchy, or just a reflection of the changing scientific interests between the 1890s and the 1990s, I couldn't say.)
- Recommended (introductory):
- Geoffrey Grimmett and David Stirzaker, Probability and Random Processes [This is my favorite probability textbook, and returns to branching processes in many places.]
- Recommended (forbiddingly technical):
- P. Del Moral and L. Miclo, "Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Nonlinear Filtering", in J. Azema, M. Emery, M. Ledoux and M. Yor (eds)., Semainaire de Probabilites XXXIV (Springer-Verlag, 2000), pp. 1--145 [Postscript preprint. Looks like a trial run for Del Moral's book, below, which I've yet to read.]
To read:
