Silviu Guiasu

The principle of maximum entropy

There is no need to stress the importance of variational problems in mathematics and its applications. The list of variational problems, of different degrees of difficulty, is very long, and it stretches from famous minimum and maximum problems of antiquity, through the variational problems of analytical mechanics and theoretical physics, all the way to the variational problems of modern opera t ions research. While maximizing or minimizing a function or a functional is a routine procedure, some special variational problems give solutions which either unify previously unconnected results or match surprisingly well the results of our experiments. Such variational problems are called variational principles. Whether or not the architecture of our world is based on variational principles is a philosophical problem. But it is a sound strategy to discover and apply variational principles in order to acquire a better understanding of a part of this architecture. In applied mathematics we get a model by taking into account some connections and, inevitably, ignoring others. One way of making a model convincing and useful is to obtain it as the solution of a variational problem. The aim of the present paper is to bring some arguments in favour of the promotion of the variational problem of entropy maximization to the rank of a variational principle.

The Weighted Gini-Simpson Index: Revitalizing an Old Index of Biodiversity

The distribution of biodiversity at multiple sites of a region has been traditionally investigated through the additive partitioning of the regional biodiversity into the average within-site biodiversity and the biodiversity among sites. The standard additive partitioning of diversity requires the use of a measure of diversity, which is a concave function of the relative abundance of species, such as the Gini-Simpson index, for instance. Recently, it was noticed that the widely used Gini-Simpson index does not behave well when the number of species is very large. The objective of this paper is to show that the new weighted Gini-Simpson index preserves the qualities of the classic Gini-Simpson index and behaves very well when the number of species is large. The weights allow us to take into account the abundance of species, the phylogenetic distance between species, and the conservation values of species. This measure may also be generalized to pairs of species and, unlike Rao’s index, this measure proves to be a concave function of the joint distribution of the relative abundance of species, being suitable for use in the additive partitioning of biodiversity. The weighted Gini-Simpson index may be easily transformed for use in the multiplicative partitioning of biodiversity as well.

Grouping data by using the weighted entropy

Abstract The aim of the paper is to show how the weighted entropy, a generalization of Shannons entropy, may be used to balance the amount of information and the degree of homogeneity associated to a partition of data in classes. Particularly, a new interpretation of Sturgess rule is given.

Conditional and weighted measures of ecological diversity

Shannons entropy and Simpsons index are the most used measures of species diversity. As the Simpson index proves to be just an approximation of the Shannon entropy, conditional Simpson indices of diversity and a global measure of interdependence among species are introduced, similar to those used in the corresponding entropic formalism from information theory. Also, since both the Shannon entropy and the Simpson index depend only on the number and relative abundance of the respective species in a given ecosystem, the paper generalizes these indices of diversity to the case when a numerical weight is attached to each species. Such a weight could reflect supplementary information about the absolute abundance, the economic significance, or the conservation value of the species.

A classification of the main probability distributions by minimizing the weighted logarithmic measure of deviation

The paper reanalyzes the following nonlinear program: Find the most similar probability distribution to a given reference measure subject to constraints expressed by mean values by minimizing the weighted logarithmic deviation. The main probability distributions are examined from this point of view and the results are summarized in a table.

The relative information generating function

Abstract The aim of the paper is to introduce the relative information generating function whose derivatives give well-known statistical indices such as the Kullback-Leibler divergence between two probability distributions and Watanabes measure of interdependence. It contains Golombs generating function as a particular case, and takes simple forms for the binomial and the Poisson distributions, neither of which fits into Golombs model. Besides its unifying role, the relative information generating function suggests new information indices as, for instance, the standard deviation of the variation of information.

Three ancient problems solved by using the game theory logic based on the Shapley value

The ancient problems of bankruptcy, contested garment, and rights arbitration have generated many studies, debates, and controversy. The objective of this paper is to show that the Shapley value from game theory, measuring the power of each player in a game, may be consistently applied for getting the general one-step solution of all these three problems viewed as n-person games. The decision making is based on the same tool, namely the game theory logic based on the use of the Shapley value, but the specific games involved are slightly different in each problem. The kind of claims of the players, the relationship between the given claims and the given resources available, and the particular way of calculating the generalized characteristic function of the game determine the specific type of game which has to be solved in each of the three ancient problems mentioned. The iterative use of the Shapley value may also justify the well-known Aumann–Maschler step-by-step procedure for solving the bankruptcy problem.

On The Principle of Minimum interdependence

Applying the Principle of Minimum Interdependence (abbreviated as PIM) we determine the largest product probability distribution compatible with some data on the interdependence between two or several random variables (some mixed moments of random variables). The general solution both in the discrete case and in the continuous one is given. A special analysis is done for the normal random variables. Particularly, the minimum interdependence between normal random variables compatible with the given covariance matrix is computed.

Coping with uncertainty in n-person games

A solution of n-person games is proposed, based on the minimum deviation from statistical equilibrium subject to the constraints imposed by the group rationality and individual rationality. The new solution is compared with the Shapley value and von Neumann-Morgensterns core of the game in the context of the 15-person game of passing and defeating resolutions in the UN Security Council involving five permanent members and ten nonpermanent members. A coalition classification, based on the minimum ramification cost induced by the characteristic function of the game, is also presented.

NEW MEASURES FOR COMPARING THE SPECIES DIVERSITY FOUND IN TWO OR MORE HABITATS

Both the weighted entropy, which generalizes the Shannon entropy, and the weighted quadratic index, which generalizes the Gini-Simpson index, are used for getting a unified treatment of some diversity measures proposed recently in ecology. The weights may reflect the ecological importance, rarity, or economic value of the species from a given habitat. The weighted measures, being concave functions, may be used in the additive partition of diversity. The weighted quadratic index has a special advantage over the weighted entropy because its maximum value has a simple analytical formula which allows us to introduce a normed measure of dissimilarity between habitats. A special case of weighted quadratic index is the Rich-Gini-Simpson index which, unlike the Shannon entropy and the classic Gini-Simpson index, behaves well when the number of species is very large. The weighted entropy and the weighted quadratic index may also be used to measure the global diversity among the subsets of species. In this context, Raos quadratic index of diversity between the pairs of species, based on the phylogenetic distance between species, is obtained as a particular case and is generalized to measure the diversity among the triads of species as well.