Modern machine learning algorithms have achieved remarkable performance in a myriad of applications, and are increasingly used to make impactful decisions in the hiring process, criminal sentencing, healthcare diagnostics and even to make new scientific discoveries. The use of data-driven algorithms in high-stakes applications is exciting yet alarming: these methods are extremely complex, often brittle, notoriously hard to analyze and interpret. Naturally, concerns have raised about the reliability, fairness, and reproducibility of the output of such algorithms.

Scientific research is often concerned with questions of cause and effect. For example, does eating processed meat cause certain types of cancer? Ideally, such questions are answered by randomized controlled experiments. However, these experiments can be costly, time-consuming, unethical or impossible to conduct. Hence, often the only available data to answer causal questions is observational.  

Change point detection is a popular tool for identifying locations in a data sequence where an abrupt change occurs in the data distribution and has been widely studied for Euclidean data. Modern data very often is non- Euclidean, for example distribution valued data or network data. Change point detection is a challenging problem when the underlying data space is a metric space where one does not have basic algebraic operations like addition of the data points and scalar multiplication. 

Advisor: Jon Wellner We consider the problem of forming confidence intervals and tests for the location of the mode in the setting of nonparametric estimation of a log-concave density. We thus study the class of log-concave densities with fixed and known mode. We find the maximum likelihood estimator for this class, give a characterization of it, and, under the null hypothesis, show our estimator is uniformly consistent and is $n^{2/5}$-tight at the mode. We also show uniqueness of the analogous limiting "estimator" of a quadratic function with white noise.

Advisor: Adrian E. Raftery