Prior design is one of the most important problems in both statistics and machine learning. The cross validation (CV) and the widely applicable information criterion (WAIC) are predictive measures of the Bayesian estimation, however, it has been difficult to apply them to find the optimal prior because their mathematical properties in prior evaluation have been unknown and the region of the hyperparameters is too wide to be examined. In this paper, we derive a new formula by which the theoretical relation among CV, WAIC, and the generalization loss is clarified and the optimal hyperparameter can be directly found. By the formula, three facts are clarified about predictive prior design. Firstly, CV and WAIC have the same second order asymptotic expansion, hence they are asymptotically equivalent to each other as the optimizer of the hyperparameter. Secondly, the hyperparameter which minimizes CV or WAIC makes the average generalization loss to be minimized asymptotically but does not the random generalization loss. And lastly, by using the mathematical relation between priors, the variances of the optimized hyperparameters by CV and WAIC are made smaller with small computational costs. Also we show that the optimized hyperparameter by DIC or the marginal likelihood does not minimize the average or random generalization loss in general.
Linear and Quadratic Discriminant analysis (LDA/QDA) are common tools for classification problems. For these methods we assume observations are normally distributed within group. We estimate a mean and covariance matrix for each group and classify using Bayes theorem. With LDA, we estimate a single, pooled covariance matrix, while for QDA we estimate a separate covariance matrix for each group. Rarely do we believe in a homogeneous covariance structure between groups, but often there is insufficient data to separately estimate covariance matrices. We propose L1- PDA, a regularized model which adaptively pools elements of the precision matrices. Adaptively pooling these matrices decreases the variance of our estimates (as in LDA), without overly biasing them. In this paper, we propose and discuss this method, give an efficient algorithm to fit it for moderate sized problems, and show its efficacy on real and simulated datasets.
We study the problem of fair binary classification using the notion of Equal Opportunity. It requires the true positive rate to distribute equally across the sensitive groups. Within this setting we show that the fair optimal classifier is obtained by recalibrating the Bayes classifier by a group-dependent threshold. We provide a constructive expression for the threshold. This result motivates us to devise a plug-in classification procedure based on both unlabeled and labeled datasets.
In order to investigate the breast cancer prediction problem on the aging population with the grades of DCIS, we conduct a tree augmented naive Bayesian network experiment trained and tested on a large clinical dataset including consecutive diagnostic mammography examinations, consequent biopsy outcomes and related cancer registry records in the population of women across all ages. The aggregated results of our ten-fold cross validation method recommend a biopsy threshold higher than 2% for the aging population.
This paper describes an effort to measure the effectiveness of tutor help in an intelligent tutoring system. Although conventional pre-and post-test experiments can determine whether tutor help is effective, they are expensive to conduct. Furthermore, pre-and post-test experiments often do not model student knowledge explicitly and thus are ignoring a source of information: students often request help about words they do not know. Therefore, we construct a dynamic Bayes net (which we call the Help model) that models tutor help and student knowledge in one coherent framework. The Help model distinguishes two different effects of help: scaffolding immediate performance vs. teaching persistent knowledge that improves long term performance. We train the Help model to fit student performance data gathered from usage of the Reading Tutor (Mostow & Aist, 2001). The parameters of the trained model suggest that students benefit from both the scaffolding and teaching effects of help. That is, students are more likely to perform correctly on the current attempt and learn persistent knowledge if tutor help is provided. Thus, our framework is able to distinguish two types of influence that tutor help has on the student, and can determine whether help helps learning without an explicit controlled study.