Bayesian analysis of systems with random chemical composition: renormalization-group approach to Dirichlet distributions and the statistical theory of dilution.
We investigate the statistical properties of systems with random chemical composition and try to obtain a theoretical derivation of the self-similar Dirichlet distribution, which is used empirically in molecular biology, environmental chemistry, and geochemistry. We consider a system made up of many chemical species and assume that the statistical distribution of the abundance of each chemical species in the system is the result of a succession of a variable number of random dilution events, which can be described by using the renormalization-group theory. A Bayesian approach is used for evaluating the probability density of the chemical composition of the system in terms of the probability densities of the abundances of the different chemical species. We show that for large cascades of dilution events, the probability density of the composition vector of the system is given by a self-similar probability density of the Dirichlet type. We also give an alternative formal derivation for the Dirichlet law based on the maximum entropy approach, by assuming that the average values of the chemical potentials of different species, expressed in terms of molar fractions, are constant. Although the maximum entropy approach leads formally to the Dirichlet distribution, it does not clarify the physical origin of the Dirichlet statistics and has serious limitations. The random theory of dilution provides a physical picture for the emergence of Dirichlet statistics and makes it possible to investigate its validity range. We discuss the implications of our theory in molecular biology, geochemistry, and environmental science.
Vlad
Marcel Ovidiu
MO
Department of Chemistry, Stanford University, Stanford, California 94305-5080, USA.
Tsuchiya
Masa
M
Oefner
Peter
P
Ross
John
J
D000465:Algorithms; D001499:Bayes Theorem; D001671:Biochemistry / Q000379:methods; D004247:DNA / Q000737:chemistry; D008956:Models, Chemical; D015233:Models, Statistical
D004247:DNA