Hypothesis Testing

PowerPoint slides for Hypothesis Testing. Examples are taken from the Medical Literatur

Active sequential hypothesis testing

Consider a decision maker who is responsible to dynamically collect observations so as to enhance his information about an underlying phenomena of interest in a speedy manner while accounting for the penalty of wrong declaration. Due to the sequential nature of the problem, the decision maker relies on his current information state to adaptively select the most ``informative'' sensing action among the available ones. In this paper, using results in dynamic programming, lower bounds for the optimal total cost are established. The lower bounds characterize the fundamental limits on the maximum achievable information acquisition rate and the optimal reliability. Moreover, upper bounds are obtained via an analysis of two heuristic policies for dynamic selection of actions. It is shown that the first proposed heuristic achieves asymptotic optimality, where the notion of asymptotic optimality, due to Chernoff, implies that the relative difference between the total cost achieved by the proposed policy and the optimal total cost approaches zero as the penalty of wrong declaration (hence the number of collected samples) increases. The second heuristic is shown to achieve asymptotic optimality only in a limited setting such as the problem of a noisy dynamic search. However, by considering the dependency on the number of hypotheses, under a technical condition, this second heuristic is shown to achieve a nonzero information acquisition rate, establishing a lower bound for the maximum achievable rate and error exponent. In the case of a noisy dynamic search with size-independent noise, the obtained nonzero rate and error exponent are shown to be maximum.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1144 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Non-Stochastic Hypothesis Testing with Application to Privacy Against Hypothesis-Testing Adversary

In this paper, we consider privacy against hypothesis testing adversaries within a non-stochastic framework. We develop a theory of non-stochastic hypothesis testing by borrowing the notion of uncertain variables from non-stochastic information theory. We define tests as binary-valued mappings on uncertain variables and prove a fundamental bound on the best performance of tests in non-stochastic hypothesis testing. We use this bound to develop a measure of privacy. We then construct reporting policies with prescribed privacy and utility guarantees. The utility of a reporting policy is measured by the distance between the reported and original values. We illustrate the effects of using such privacy-preserving reporting polices on a publicly-available practical dataset of preferences and demographics of young individuals, aged between 15-30, with Slovakian nationality

Universal Outlier Hypothesis Testing

Outlier hypothesis testing is studied in a universal setting. Multiple sequences of observations are collected, a small subset of which are outliers. A sequence is considered an outlier if the observations in that sequence are distributed according to an ``outlier'' distribution, distinct from the ``typical'' distribution governing the observations in all the other sequences. Nothing is known about the outlier and typical distributions except that they are distinct and have full supports. The goal is to design a universal test to best discern the outlier sequence(s). It is shown that the generalized likelihood test is universally exponentially consistent under various settings. The achievable error exponent is also characterized. In the other settings, it is also shown that there cannot exist any universally exponentially consistent test.Comment: IEEE Trans. Inf. Theory, to appear, 201

Differentially Private Nonparametric Hypothesis Testing

Hypothesis tests are a crucial statistical tool for data mining and are the workhorse of scientific research in many fields. Here we study differentially private tests of independence between a categorical and a continuous variable. We take as our starting point traditional nonparametric tests, which require no distributional assumption (e.g., normality) about the data distribution. We present private analogues of the Kruskal-Wallis, Mann-Whitney, and Wilcoxon signed-rank tests, as well as the parametric one-sample t-test. These tests use novel test statistics developed specifically for the private setting. We compare our tests to prior work, both on parametric and nonparametric tests. We find that in all cases our new nonparametric tests achieve large improvements in statistical power, even when the assumptions of parametric tests are met