Hamed Masnadi-Shirazi

Hamed Masnadi-Shirazi

The two-sample hypothesis testing problem is studied for the challenging scenario of high dimensional data sets with small sample sizes. We show that the two-sample hypothesis testing problem can be posed as a one-class set classification problem. In the set classification problem the goal is to classify a set of data points that are assumed to have a common class. We prove that the average probability of error given a set is less than or equal to the Bayes error and decreases as a power of $n$ number of sample data points in the set. We use the positive definite Set Kernel for directly mapping sets of data to an associated Reproducing Kernel Hilbert Space, without the need to learn a probability distribution. We specifically solve the two-sample hypothesis testing problem using a one-class SVM in conjunction with the proposed Set Kernel. We compare the proposed method with the Maximum Mean Discrepancy, F-Test and T-Test methods on a number of challenging simulated high dimensional and small sample size data. We also perform two-sample hypothesis testing experiments on six cancer gene expression data sets and achieve zero type-I and type-II error results on all data sets.

Hamed Masnadi-Shirazi

We study strictly proper scoring rules in the Reproducing Kernel Hilbert Space. We propose a general Kernel Scoring rule and associated Kernel Divergence. We consider conditions under which the Kernel Score is strictly proper. We then demonstrate that the Kernel Score includes the Maximum Mean Discrepancy as a special case. We also consider the connections between the Kernel Score and the minimum risk of a proper loss function. We show that the Kernel Score incorporates more information pertaining to the projected embedded distributions compared to the Maximum Mean Discrepancy. Finally, we show how to integrate the information provided from different Kernel Divergences, such as the proposed Bhattacharyya Kernel Divergence, using a one-class classifier for improved two-sample hypothesis testing results.

Hamed Masnadi-Shirazi

The concept of refinement from probability elicitation is considered for proper scoring rules. Taking directions from the axioms of probability, refinement is further clarified using a Hilbert space interpretation and reformulated into the underlying data distribution setting where connections to maximal marginal diversity and conditional entropy are considered and used to derive measures that provide arbitrarily tight bounds on the Bayes error. Refinement is also reformulated into the classifier output setting and its connections to calibrated classifiers and proper margin losses are established.