Closures of exponential families

The variation distance closure of an exponential family with a convex set of canonical parameters is described, assuming no regularity conditions. The tools are the concepts of convex core of a measure and extension of an exponential family, introduced previously by the authors, and a new concept of accessible faces of a convex set. Two other closures related to the information divergence are also characterized.Comment: Published at http://dx.doi.org/10.1214/009117904000000766 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Entropy Region and Convolution

The entropy region is constructed from vectors of random variables by collecting Shannon entropies of all subvectors. Its shape is studied here by means of polymatroidal constructions, notably by convolution. The closure of the region is decomposed into the direct sum of tight and modular parts, reducing the study to the tight part. The relative interior of the reduction belongs to the entropy region. Behavior of the decomposition under self-adhesivity is clarified. Results are specialized and extended to the region constructed from four tuples of random variables. This and computer experiments help to visualize approximations of a symmetrized part of the entropy region. The four-atom conjecture on the minimal Ingleton score is refuted. © 2016 IEEE

Convergence of generalized entropy minimizers in sequences of convex problems

Integral functionals based on convex normal integrands are minimized over convex constraint sets. Generalized minimizers exist under a boundedness condition. Sequences of the minimization problems are studied when the constraint sets are nested. The corresponding sequences of generalized minimizers are related to the minimization over limit convex sets. Martingale theorems and moment problems are discussed. © 2016 IEEE

Extreme Convex Set Functions With Many Nonnegative Differences

. Where N is a finite set of the cardinality n and P the family of all its subsets, we study real functions on P having nonnegative differences of orders n \Gamma 2, n \Gamma 1 and n. Nonnegative differences of zeroth order, first order, and second order may be interpreted as nonnegativity, nonincreasingness and convexity, respectively. If all differences up to order n of a function are nonnegative, the set function is called completely monotone in analogy to the continuous case. We present a discrete Bernstein-type theorem for these functions with Mobius inversion in the place of Laplace one. Numbers of all extreme functions with nonnegative differences up to the orders n, n \Gamma 1 and n \Gamma 2, which is the most sophisticated case, and their Mobius transforms are found. As an example, we write out all extreme nonnegative nondecreasing and semimodular functions to the set N with four elements. 1. Introduction Let N be a finite basic set of the cardinality n = jN j and P s r (N ..

Probabilistic Conditional Independence Structures And Matroid Theory: Background

. Special conditional independence structures have been recognized to be matroids. This opens new possibilities for the application of matroid theory methods (duality, minors, expansions) to the study of conditional independence and, on the other hand, starts a new probabilistic branch of matroid representation theory. 2 1. INTRODUCTION Though of old origin, the concept of conditional stochastic independence has been reattracting attention of mathematicians during last decades. A new viewpoint, from Dawid (1979), consists in the simultaneous examination of all conditional independences (among triples of subsystems of a stochastic system) separately from the joint probability distribution. Subsequent development of this idea has been influenced by the logic of integrity constraints from databases (see Pearl(1988), Geiger, Pearl(1989), Studen'y(1992)) and is oriented toward searching for plausible conditional independence relations (Oliver, Smith(1990), Mat'us(1992a)). In the thirties s..

Generalized Maximum Likelihood Estimates for Infinite Dimensional Exponential Families

The notion of generalized maximum likelihood estimate for finite dimensional canonically convex exponential families, studied in detail in previous works of the authors, is extended to an infinite dimensional setting. Existence of the estimate when a generalized log-likelihood function is bounded above, and a continuity property are established. Related literature and examples are discussed

Engineering Applications of Artificial Intelligence in Mechanical Design and Optimization

This study offers a complete analysis of the use of deep learning or machine learning, as well as precise recommendations on how these methods could be used in the creation of machine components and nodes. The examples in this thesis are intended to identify areas in mechanical design and optimization where this technique could be widely applied in the future, benefiting society and advancing the current state of modern mechanical engineering. The review begins with a discussion on the workings of artificial intelligence, machine learning, and deep learning. Different techniques, classifications, and even comparisons of each method are described in detail. The most common programming languages, frameworks, and software used in mechanical engineering for this problem are gradually introduced. Input data formats and the most common datasets that are suitable for the field of machine learning in mechanical design and optimization are also discussed. The second half of the review describes the current use of machine learning in several areas of mechanical design and optimization, using specific examples that have been investigated by researchers from around the world. Further research directions on the use of machine learning and neural networks in the fields of mechanical design and optimization are discussed

Investigating the Spectrum of Biological Activity of Ring-Substituted Salicylanilides and Carbamoylphenylcarbamates

In this study, a series of twelve ring-substituted salicylanilides and carbamoylphenylcarbamates were prepared and characterized. The compounds were analyzed using RP-HPLC to determine lipophilicity. They were tested for their activity related to the inhibition of photosynthetic electron transport (PET) in spinach (Spinacia oleracea L.) chloroplasts. Moreover, their site of action in the photosynthetic apparatus was determined. Primary in vitro screening of the synthesized compounds was also performed against mycobacterial, bacterial and fungal strains. Several compounds showed biological activity comparable with or higher than the standards 3-(3,4-dichlorophenyl)-1,1-dimethylurea, isoniazid, penicillin G, ciprofloxacin or fluconazole. The most active compounds showed minimal anti-proliferative activity against human cells in culture, indicating they would have low cytotoxicity. For all compounds, the relationships between lipophilicity and the chemical structure are discussed