This monograph presents a comprehensive treatment of the maximum-entropy sampling problem (MESP), which is a fascinating topic at the intersection of mathematical optimization and data science. The text situates MESP in information theory, as the algorithmic problem of calculating a sub-vector of pre-specificed size from a multivariate Gaussian random vector, so as to maximize Shannon's differential entropy. The text collects and expands on state-of-the-art algorithms for MESP, and addresses its application in the field of environmental monitoring. While MESP is a central optimization problem in the theory of statistical designs (particularly in the area of spatial monitoring), this book largely focuses on the unique challenges of its algorithmic side. From the perspective of mathematical-optimization methodology, MESP is rather unique (a 0/1 nonlinear program having a nonseparable objective function), and the algorithmic techniques employed are highly non-standard. In particular, successful techniques come from several disparate areas within the field of mathematical optimization; for example: convex optimization and duality, semidefinite programming, Lagrangian relaxation, dynamic programming, approximation algorithms, 0/1 optimization (e.g., branch-and-bound), extended formulation, and many aspects of matrix theory. The book is mainly aimed at graduate students and researchers in mathematical optimization and data analytics.

Given a set of randomly sampled experimental data, a numeric algorithm has been developed to determine a probability distribution function that represents the data. The algorithm is based on the maximum entropy method in information theory, which states that the solution that maximizes entropy while satisfying known constraints is presumed to be the correct solution. The probability distribution function is expressed as an exponential of a series expansion over Chebyshev polynomials where the expansion coefficients are formally Lagrange multipliers and must be determined. The method for finding these Lagrange multipliers is through an iterative trial-and-error approach. After each guess for the coefficients, the trial solution is evaluated according to how well it represents the data. This assessment is performed by using the cumulative distribution function to transform the data to uniformly distributed random data, and then comparing the transformed data to known results for a uniform distribution using binning and ordered statistics. The algorithm is presented in the form of a highly adaptive and flexible tool that does not assume prior knowledge about the data, but provides many documented settings to allow for maximum efficiency in the event the user has more information about the data or has specific requirements. To demonstrate the power and flexibility of this approach, it has been tested and compared against known distributions, many of which have specific properties known to cause standard methods to fail. Each of these examples demonstrates a convergence to the true function as the sample size increases.

Scientific knowledge grows at a phenomenal pace--but few books have had as lasting an impact or played as important a role in our modern world as The Mathematical Theory of Communication, published originally as a paper on communication theory more than fifty years ago. Republished in book form shortly thereafter, it has since gone through four hardcover and sixteen paperback printings. It is a revolutionary work, astounding in its foresight and contemporaneity. The University of Illinois Press is pleased and honored to issue this commemorative reprinting of a classic.

The MaxEnt workshops are devoted to Bayesian inference and maximum entropy methods in science and engineering. In addition, this workshop included all aspects of probabilistic inference, such as foundations, techniques, algorithms, and applications. All papers have been peer-reviewed.

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