### Exact Exponent in Optimal Rates for Crowdsourcing

In many machine learning applications, crowdsourcing has become the primary means for label collection. In this paper, we study the optimal error rate for aggregating labels provided by a set of non-expert workers. Under the classic Dawid-Skene model, we establish matching upper and lower bounds with an exact exponent $mI(\pi)$ in which $m$ is the number of workers and $I(\pi)$ the average Chernoff information that characterizes the workers' collective ability. Such an exact characterization of the error exponent allows us to state a precise sample size requirement $m>\frac{1}{I(\pi)}\log\frac{1}{\epsilon}$ in order to achieve an $\epsilon$ misclassification error. In addition, our results imply the optimality of various EM algorithms for crowdsourcing initialized by consistent estimators.

### Extensions of a Theory of Networks for Approximation and Learning: Outliers and Negative Examples

Bruno Caprile I.R.S.T. Povo, Italy, 38050 Learning an input-output mapping from a set of examples can be regarded as synthesizing an approximation of a multidimensional function.

### WS06-05-006.pdf

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.

### A tree augmented naive Bayesian network experiment for breast cancer prediction

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.

### Audits as Evidence: Experiments, Ensembles, and Enforcement

We develop tools for utilizing correspondence experiments to detect illegal discrimination by individual employers. Employers violate US employment law if their propensity to contact applicants depends on protected characteristics such as race or sex. We establish identification of higher moments of the causal effects of protected characteristics on callback rates as a function of the number of fictitious applications sent to each job ad. These moments are used to bound the fraction of jobs that illegally discriminate. Applying our results to three experimental datasets, we find evidence of significant employer heterogeneity in discriminatory behavior, with the standard deviation of gaps in job-specific callback probabilities across protected groups averaging roughly twice the mean gap. In a recent experiment manipulating racially distinctive names, we estimate that at least 85% of jobs that contact both of two white applications and neither of two black applications are engaged in illegal discrimination. To assess the tradeoff between type I and II errors presented by these patterns, we consider the performance of a series of decision rules for investigating suspicious callback behavior under a simple two-type model that rationalizes the experimental data. Though, in our preferred specification, only 17% of employers are estimated to discriminate on the basis of race, we find that an experiment sending 10 applications to each job would enable accurate detection of 7-10% of discriminators while falsely accusing fewer than 0.2% of non-discriminators. A minimax decision rule acknowledging partial identification of the joint distribution of callback rates yields higher error rates but more investigations than our baseline two-type model. Our results suggest illegal labor market discrimination can be reliably monitored with relatively small modifications to existing audit designs.