In real world applications, graphical statistical models are not only a tool for operations such as classification or prediction, but usually the network structures of the models themselves are also of great interest (e.g., in modeling brain connectivity). The false discovery rate (FDR), the expected ratio of falsely claimed connections to all those claimed, is often a reasonable error-rate criterion in these applications. However, current learning algorithms for graphical models have not been adequately adapted to the concerns of the FDR. The traditional practice of controlling the type I error rate and the type II error rate under a conventional level does not necessarily keep the FDR low, especially in the case of sparse networks. In this paper, we propose embedding an FDR-control procedure into the PC algorithm to curb the FDR of the skeleton of the learned graph. We prove that the proposed method can control the FDR under a user-specified level at the limit of large sample sizes. In the cases of moderate sample size (about several hundred), empirical experiments show that the method is still able to control the FDR under the user-specified level, and a heuristic modification of the method is able to control the FDR more accurately around the user-specified level. The proposed method is applicable to any models for which statistical tests of conditional independence are available, such as discrete models and Gaussian models.

During my past two years of research, I had the chance to meet and collaborate with many great people, and have been constantly supported by many people whom I cherish dearly. I am pleased to express my gratitude to all of them for making this thesis possible. First and foremost, I would like to thank my advisor, professor Joelle Pineau, for her support and insightful comments and suggestions throughout this work. My regular meetings with her were always enjoyable and fruitful. She let me explore my own ideas, while keeping me on track and providing invaluable encouragements and advice through difficult times. Her guidance helped me improve my research and writing skills, and allowed me to become a better researcher. I am grateful for her help co-writing and editing this thesis and several other research papers. I would also like to thank my thesis external examiner, professor Michael Bowling, for his time spent reviewing this thesis and for his constructive comments, which helped improve the quality of this thesis. Special thanks go to Brahim Chaib-draa, my advisor as an undergrad, for giving

If I can attain half of the success you have achieved in marriage and in life, I will have lived a full and purposeful life. A son could not ask for better parents. ii ACKNOWLEDGEMENTS I am extremely grateful to have been advised by a brilliantly creative human being. Professor Alfred Hero has allowed me to mature as an independent researcher capable of abstractly analyzing complex problems. Between day to day interactions, coursework, and discussions about research, I am forever grateful for the interactions I have had with my committee members Professors Burns, Burmeister, Shedden, and Zhu. I will always appreciate the hands on interaction and development of ideas with my post-doctoral researchers Mark Kliger and Ami Wiesel. I am so appreciative of the time and effort you both spent with me, especially early on in my graduate student career. I cannot thank Arvind Rao enough for his wisdom and insight into developing good research topics and sharing a few good laughs on our road trip to Madison, WI. I know he will be a very successful faculty member some day. For both academic collaboration and extracurricular mischief, I will never forget the moments spent with fellow graduate students and now life long friends Yongsheng Huang and Arnau Tibau Puig. Most importantly, I thank my beautiful wife and best friend Erica for her love and patience over these past four years while pursuing my Ph.D. iii TABLE OF CONTENTS

We study the problem of learning Bayesian network structures from data. We develop an algorithm for finding the k-best Bayesian network structures. We propose to compute the posterior probabilities of hypotheses of interest by Bayesian model averaging over the k-best Bayesian networks. We present empirical results on structural discovery over several real and synthetic data sets and show that the method outperforms the model selection method and the stateof-the-art MCMC methods. 1