Maximum Entropy Distribution of Correlated Variables: Application to Human Height and Weight

Abstract

Recent investigations have shown that a bivariant lognormal probability density function predicts the statistical moments and correlations of adult human height and weight so extensively and closely as to pose an enigma regarding the underlying reason for such exactness. No genetic or environmental cause currently accounts for this distribution. In this article, it is shown that the Principle of Maximum Entropy (PME)—which is an inferential method drawn exclusively from probability theory—leads to the joint lognormal distribution of height and weight independent of any physical mechanism of biological development. The operation of the PME entails carrying out a variational procedure on a functional comprising the Shannon information entropy subject to constraints posed by known prior information in the form of expectation values. In the case of height and weight the prior information consists of the means, variances, and linear correlation of the logarithms of the two variables. When applied to a large anthropometric survey, the maximum entropy distribution resulting from this variational procedure is shown to be astronomically more probable than any other distribution consistent with the prior information. Although the PME provides an explanation of the enigma, the possibility is examined that an underlying stochastic mechanism may also lead to the same distribution.