Reproducing Kernel Hilbert Space (RKHS) embedding of probability distributions has proved to be an effective approach, via MMD (maximum mean discrepancy) for nonparametric hypothesis testing problems involving distributions defined over general (non-Euclidean) domains. While a substantial amount of work has been done on this topic, only recently, minimax optimal two-sample tests have been constructed that incorporate, unlike MMD, both the mean element and a regularized version of the covariance operator. However, as with most kernel algorithms, the computational complexity of the optimal test scales cubically in the sample size, limiting its applicability. In this paper, we propose a spectral regularized two-sample test based on random Fourier feature (RFF) approximation and investigate the trade-offs between statistical optimality and computational efficiency. We show the proposed test to be minimax optimal if the approximation order of RFF (which depends on the smoothness of the likelihood ratio and the decay rate of the eigenvalues of the integral operator) is sufficiently large. We develop a practically implementable permutation-based version of the proposed test with a data-adaptive strategy for selecting the regularization parameter and the kernel. Finally, through numerical experiments on simulated and benchmark datasets, we demonstrate that the proposed RFF-based test is computationally efficient and performs almost similar (with a small drop in power) to the exact test.

Assessment of proficiency of the learner is an essential part of Intelligent Tutoring Systems (ITS). We use Item Response Theory (IRT) in computer-aided language learning for assessment of student ability in two contexts: in test sessions, and in exercises during practice sessions. Exhaustive testing across a wide range of skills can provide a detailed picture of proficiency, but may be undesirable for a number of reasons. Therefore, we first aim to replace exhaustive tests with efficient but accurate adaptive tests. We use learner data collected from exhaustive tests under imperfect conditions, to train an IRT model to guide adaptive tests. Simulations and experiments with real learner data confirm that this approach is efficient and accurate. Second, we explore whether we can accurately estimate learner ability directly from the context of practice with exercises, without testing. We transform learner data collected from exercise sessions into a form that can be used for IRT modeling. This is done by linking the exercises to {\em linguistic constructs}; the constructs are then treated as "items" within IRT. We present results from large-scale studies with thousands of learners. Using teacher assessments of student ability as "ground truth," we compare the estimates obtained from tests vs. those from exercises. The experiments confirm that the IRT models can produce accurate ability estimation based on exercises.

A test is adaptive when its sequence and number of questions is dynamically tuned on the basis of the estimated skills of the taker. Graphical models, such as Bayesian networks, are used for adaptive tests as they allow to model the uncertainty about the questions and the skills in an explainable fashion, especially when coping with multiple skills. A better elicitation of the uncertainty in the question/skills relations can be achieved by interval probabilities. This turns the model into a credal network, thus making more challenging the inferential complexity of the queries required to select questions. This is especially the case for the information theoretic quantities used as scores to drive the adaptive mechanism. We present an alternative family of scores, based on the mode of the posterior probabilities, and hence easier to explain. This makes considerably simpler the evaluation in the credal case, without significantly affecting the quality of the adaptive process. Numerical tests on synthetic and real-world data are used to support this claim.

In identifying infected patients in a population, group testing is an effective method to reduce the number of tests and correct the test errors. In the group testing procedure, tests are performed on pools of specimens collected from patients, where the number of pools is lower than that of patients. The performance of group testing heavily depends on the design of pools and algorithms that are used in inferring the infected patients from the test outcomes. In this paper, an adaptive design method of pools based on the predictive distribution is proposed in the framework of Bayesian inference. The proposed method executed using the belief propagation algorithm results in more accurate identification of the infected patients, as compared to the group testing performed on random pools determined in advance.

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