Universal and Composite Hypothesis Testing via Mismatched Divergence

For the universal hypothesis testing problem, where the goal is to decide between the known null hypothesis distribution and some other unknown distribution, Hoeffding proposed a universal test in the nineteen sixties. Hoeffding's universal test statistic can be written in terms of Kullback-Leibler (K-L) divergence between the empirical distribution of the observations and the null hypothesis distribution. In this paper a modification of Hoeffding's test is considered based on a relaxation of the K-L divergence test statistic, referred to as the mismatched divergence. The resulting mismatched test is shown to be a generalized likelihood-ratio test (GLRT) for the case where the alternate distribution lies in a parametric family of the distributions characterized by a finite dimensional parameter, i.e., it is a solution to the corresponding composite hypothesis testing problem. For certain choices of the alternate distribution, it is shown that both the Hoeffding test and the mismatched test have the same asymptotic performance in terms of error exponents. A consequence of this result is that the GLRT is optimal in differentiating a particular distribution from others in an exponential family. It is also shown that the mismatched test has a significant advantage over the Hoeffding test in terms of finite sample size performance. This advantage is due to the difference in the asymptotic variances of the two test statistics under the null hypothesis. In particular, the variance of the K-L divergence grows linearly with the alphabet size, making the test impractical for applications involving large alphabet distributions. The variance of the mismatched divergence on the other hand grows linearly with the dimension of the parameter space, and can hence be controlled through a prudent choice of the function class defining the mismatched divergence.Comment: Accepted to IEEE Transactions on Information Theory, July 201

Statistical SVMs for robust detection, supervised learning, and universal classification

The support vector machine (SVM) has emerged as one of the most popular approaches to classification and supervised learning. It is a flexible approach for solving the problems posed in these areas, but the approach is not easily adapted to noisy data in which absolute discrimination is not possible. We address this issue in this paper by returning to the statistical setting. The main contribution is the introduction of a statistical support vector machine (SSVM) that captures all of the desirable features of the SVM, along with desirable statistical features of the classical likelihood ratio test. In particular, we establish the following: (i) The SSVM can be designed so that it forms a continuous function of the data, yet also approximates the potentially discontinuous log likelihood ratio test. (ii) Extension to universal detection is developed, in which only one hypothesis is labeled (a semi-supervised learning problem). (iii) The SSVM generalizes the robust hypothesis testing problem based on a moment class. Motivation for the approach and analysis are each based on ideas from information theory. A detailed performance analysis is provided in the special case of i.i.d. observations. This research was partially supported by NSF under grant CCF 07-29031, by UTRC, Motorola, and by the DARPA ITMANET program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF, UTRC, Motorola, or DARPA. I

On thresholds for robust goodness-of-fit tests

Abstract—Goodness-of-fit tests are statistical procedures used to test the hypothesis H0 that a set of observations were drawn according to some given probability distri-bution. Decision thresholds used in goodness-of-fit tests are typically set for guaranteeing a target false-alarm probability. In many popular testing procedures results on the weak convergence of the test statistics are used for setting approximate thresholds when exact computation is infeasible. In this work, we study robust procedures for goodness-of-fit where accurate models are not available for the distribution of the observations under hypothesis H0. We develop procedures for setting thresholds in two specific examples- a robust version of the Kolmogorov-Smirnov test for continuous alphabets and a robust version of the Hoeffding test for finite alphabets. I

A photochemical method for immobilization of azidated dextran onto aminated poly(ethylene terephthalate) surfaces

Background: Dextran, a bacterial polysaccharide, has been reported to be as good as poly(ethylene glycol) in its protein-rejecting and cell-repelling abilities. In addition, the multivalent nature of dextran is advantageous for surface grafting of biologically active molecules. We report here a method to photochemically bind dextran hydrogel films to aminated poly(ethylene terephthalate) (PET) surfaces in aqueous media using a heterobifunctional crosslinker, 4-azidobenzoic acid. In order to achieve this, dextran was first functionalized with the crosslinker using carbodiimide chemistry followed by photo-crosslinking and immobilization onto the nucleophile-rich aminated PET surfaces. Results: The presence of the immobilized dextran on PET was verified by attenuated total-reflection Fourier transform infrared spectroscopy, X-ray photoelectron spectroscopy, scanning electron microscopy and contact angle measurements. The grafted surface was highly hydrophilic due to the heavily hydrated polysaccharide network on the surface as demonstrated by the near zero water contact angle. Conclusion: A photochemical method for surface immobilization of dextran onto aminated PET using aryl azide chemistry is a facile technique to generate highly hydrophilic and more hemocompatible surfaces. The aryl nitrenes generated by photolysis produce a metastable, electron-deficient intermediate, azacycloheptatetraene, which is believed to be responsible for the simultaneous crosslinking of dextran and its immobilization onto the aminated PET surface. The aryl azide chemistry reported here for dextran could be useful as a versatile technique for surface modification of other nucleophile-rich polymers to create dextran- or similar polysaccharide-immobilized surfaces

Evaluating the usefulness of 50 millesimal potencies in the treatment of chronic diseases - A retrospective study

Introduction: The 50 millesimal potency, is not fully utilized in our day to day practice. This retrospective study was done to reveal a new horizon for the physicians who use it occasionally and an eye opener for those who have never tried it. Aim: The aim was to evaluate the usefulness of 50 Millesimal potency of indicated medicine in the treatment of chronic diseases from a retrospective study. Materials and Methods: Cases treated with 50 Millesimal potency (LM) during January-May 2014, were screened and based on eligibility criteria, 50 cases were selected to study retrospectively. Treatment outcome was analyzed based on follow-up criteria. Data were statistically analyzed with Chi-square test in GNU PSPP Software. Results: 50 Millesimal potencies have the potential to give significant improvement (P = 0.01) in the treatment of chronic diseases. There were no cases reported with aggravation. The action of LM potency is not influenced (P = 0.97) by previously used Centesimal potency. Constitutional prescription has proved to have significant (P = 0.01) association with treatment outcome with LM potency, whereas Sector prescription (P = 0.12) does not. Irrespective of age, gender, and duration of illness, 50 Millesimal potencies act advantageously. Conclusion: The data suggest that 50 Millesimal potencies have significant beneficial effects in the treatment of chronic diseases