Abstract
In the first part of the paper we work out the consequences of the fact that Jaynes’ Maximum Entropy Principle, when translated in mathematical terms, is a constrained extremum problem for an entropy function expressing the uncertainty associated with the probability distribution p. Consequently, if two observers use different independent variables p or , the associated entropy functions have to be defined accordingly and they are different in the general case. In the second part we apply our findings to an analysis of the foundations of the Maximum Entropy Theory of Ecology (M.E.T.E.) a purely statistical model of an ecological community. Since the theory has received considerable attention by the scientific community, we hope to give a useful contribution to the same community by showing that the procedure of application of MEP, in the light of the theory developed in the first part, suffers from some incongruences. We exhibit an alternative formulation which is free from these limitations and that gives different results.
Highlights
It is based on: (i) the enumeration of the system states i = 1, . . . , N; (ii) the introduction of one or more functions that translate the information available of the system in the form of constraints on the probability i.e., as f ( p) = c where c is a vector of average values; and (iii) a function measuring the uncertainty associated to a probability distribution candidate to describe the system
We have shown that in the case where the probability distribution being used can be linked at least conceptually to independent repetitions of an experiment, we have a criterion to derive the correct form of the entropy function from a first principle
A non trivial task in the application of the MEP is the translation in mathematical terms of the information available on the system
Summary
Maximum Entropy Principle MEP (E.T. Jaynes, 1957, see [1,2]) is a powerful inference principle which allows to determine the probability distribution that describes a system on the basis of the information available, usually in the form of averages of observables (random variables) of interest for the system. Proposition 1 in Section 2 below is the main tool to determine the form of the uncertainty function using g-related distributions The analysis of this subtile point of the application of MEP is the main aim of this paper. Using the Maximum Entropy Principle the theory aims at inferring the form of some of the most used distributions in ecology from the knowledge of macroscopic information on the system: number of individuals, number of species and total metabolic requirement.
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