@article{2010, abstract = {Many algorithms for inferring causality rely heavily on the faithfulness assumption. The main justification for imposing this assumption is that the set of unfaithful distributions has Lebesgue measure zero, since it can be seen as a collection of hypersurfaces in a hypercube. However, due to sampling error the faithfulness condition alone is not sufficient for statistical estimation, and strong-faithfulness has been proposed and assumed to achieve uniform or high-dimensional consistency. In contrast to the plain faithfulness assumption, the set of distributions that is not strong-faithful has nonzero Lebesgue measure and in fact, can be surprisingly large as we show in this paper. We study the strong-faithfulness condition from a geometric and combinatorial point of view and give upper and lower bounds on the Lebesgue measure of strong-faithful distributions for various classes of directed acyclic graphs. Our results imply fundamental limitations for the PC-algorithm and potentially also for other algorithms based on partial correlation testing in the Gaussian case.}, author = {Uhler, Caroline and Raskutti, Garvesh and Bühlmann, Peter and Yu, Bin}, journal = {The Annals of Statistics}, number = {2}, pages = {436 -- 463}, publisher = {Institute of Mathematical Statistics}, title = {{Geometry of the faithfulness assumption in causal inference}}, doi = {10.1214/12-AOS1080}, volume = {41}, year = {2013}, } @article{2009, abstract = {Traditional statistical methods for confidentiality protection of statistical databases do not scale well to deal with GWAS databases especially in terms of guarantees regarding protection from linkage to external information. The more recent concept of differential privacy, introduced by the cryptographic community, is an approach which provides a rigorous definition of privacy with meaningful privacy guarantees in the presence of arbitrary external information, although the guarantees may come at a serious price in terms of data utility. Building on such notions, we propose new methods to release aggregate GWAS data without compromising an individual’s privacy. We present methods for releasing differentially private minor allele frequencies, chi-square statistics and p-values. We compare these approaches on simulated data and on a GWAS study of canine hair length involving 685 dogs. We also propose a privacy-preserving method for finding genome-wide associations based on a differentially-private approach to penalized logistic regression.}, author = {Uhler, Caroline and Slavkovic, Aleksandra and Fienberg, Stephen}, journal = {Journal of Privacy and Confidentiality }, number = {1}, pages = {137 -- 166}, publisher = {Carnegie Mellon University}, title = {{Privacy-preserving data sharing for genome-wide association studies}}, doi = {10.29012/jpc.v5i1.629}, volume = {5}, year = {2013}, } @article{2280, abstract = {The problem of packing ellipsoids of different sizes and shapes into an ellipsoidal container so as to minimize a measure of overlap between ellipsoids is considered. A bilevel optimization formulation is given, together with an algorithm for the general case and a simpler algorithm for the special case in which all ellipsoids are in fact spheres. Convergence results are proved and computational experience is described and illustrated. The motivating application-chromosome organization in the human cell nucleus-is discussed briefly, and some illustrative results are presented.}, author = {Uhler, Caroline and Wright, Stephen}, journal = {SIAM Review}, number = {4}, pages = {671 -- 706}, publisher = {Society for Industrial and Applied Mathematics }, title = {{Packing ellipsoids with overlap}}, doi = {10.1137/120872309}, volume = {55}, year = {2013}, } @article{2959, abstract = {We study maximum likelihood estimation in Gaussian graphical models from a geometric point of view. An algebraic elimination criterion allows us to find exact lower bounds on the number of observations needed to ensure that the maximum likelihood estimator (MLE) exists with probability one. This is applied to bipartite graphs, grids and colored graphs. We also study the ML degree, and we present the first instance of a graph for which the MLE exists with probability one, even when the number of observations equals the treewidth.}, author = {Uhler, Caroline}, journal = {Annals of Statistics}, number = {1}, pages = {238 -- 261}, publisher = {Institute of Mathematical Statistics}, title = {{Geometry of maximum likelihood estimation in Gaussian graphical models}}, doi = {10.1214/11-AOS957}, volume = {40}, year = {2012}, }