Hybrid Bayesian networks for reasoning about complex systems

Many real-world systems are naturally modeled as hybrid stochastic processes, i.e., stochastic processes that contain both discrete and continuous variables. Examples include speech recognition, target tracking, and monitoring of physical systems. The task is usually to perform probabilistic inference, i.e., infer the hidden state of the system given some noisy observations. For example, we can ask what is the probability that a certain word was pronounced given the readings of our microphone, what is the probability that a submarine is trying to surface given our sonar data, and what is the probability of a valve being open given our pressure and ow readings. Bayesian networks are a compact way to represent a probability distribution. They can be extended to dynamic Bayesian networks which represent stochastic processes. In this thesis we concentrate on hybrid (dynamic) Bayesian networks. Our contributions are three-fold: theoretical, algorithmic, and practical. From a theoretical perspective, we provide a novel complexity analysis for inference in hybrid models and show that there is a fundamental di erence between the complexity of inference in discrete models and in hybrid ones. In particular, we provide the rst NP-hardness results for inference in very simple hybrid models. From an algorithmic perspective, we provide a suite of new inference algorithms, designed to deal with non-linearities present in many real-world systems, and to scale up to large hybrid models. We show that our algorithms often outperform the current state of the art inference algorithms for hybrid models. Finally, from a practical perspective, we apply our techniques to the task of fault diagnosis in a complex real-world physical system, designed to extract oxygen from the Martian atmosphere. We demonstrate the feasibility of our approach using data collected during actual runs of the system. v Acknowledgments First and foremost I would like to thank Daphne Koller. If I had to describe Daphne in one word, excellence would be my choice. Her uncompromising standards make everybody, and mostly herself, work harder, but also bring her students to ful ll their true potential. Looking back I realize how little I knew when I came to Stanford and how much I learned from Daphne during these years: how to nd interesting research problems, how to clearly present one's work both in papers and in talks, how to teach a class, and in short how to be a good researcher. I am most grateful to Daphne for bringing me to the point of being proud of my work. Excluding Daphne, Ron Parr is the person to whom I owe my biggest debt of gratitude. Ron, who was a Research Associate in Daphne's research group, served as my uno cial second adviser during a crucial period of my Ph.D. career, and to a large extent he is responsible for putting me on the track which ultimately led to this thesis. Working closely with Ron was one of the most rewarding and enjoyable experiences of my Ph.D. career. Ron always had the time for a chat, and much of my work can be traced back to these chats. I feel extremely fortunate to have met Ron and to count him as a friend. Thank you Ron! I would also like to thank Stephen Boyd, the third member of my reading committee, who pointed out to me many interesting directions and connections that turned out to be very useful. I thank Daphne, Ron and Stephen for taking the time to read this long thesis and come up with as many useful comments as they have had. I also thank Ross Shachter and Nils Nilsson who were in my Orals committee. Working on the RWGS was truly a group e ort, and I am privileged to have worked with such an amazing group of people. Among the people at Stanford I vi would like to thank Brooks Moses who was the only one of us who truly understood the system and was able to come up with a model for it, Sheila McIlraith who brought the problem into my attention and was instrumental in keeping the project on the right track, and Maricia Scott who shared much of the workload with me and was always invaluable during the crucial moments. Brooks, Maricia and I spent many hours on constructing and debugging the model. Their company transformed the experience from a potentially painful one to an enjoyable one. On the NASA side I would rst like to thank Charlie Goodrich whose help went above and beyond any reasonable expectation. Charlie was always there to patiently answer our questions, make sure we have access to the data that we needed, and was even kind enough to host us at his home during our visit to Kennedy Space Center. I could not have hoped for a more helpful or a nicer person than Charlie. I would also like to thank Dan Clancy, Bill Larson, Jon Whitlow, Clyde Parrish, Curtis Ihl eld and Dan Keenan for their help. Zohar Manna was my adviser during my rst year at Stanford, and made my transition from Israel to Stanford much smoother and easier. Zohar let me develop my research interests during my rst year, was always there when I needed someone to talk with, and always believed in my abilities. For that I am truly grateful. Shimon Ullman from the Weizmann Institute of Science in Israel helped me a great deal in my application process and encouraged me to come to Stanford. Since then he was always there to keep track of my progress and give me a good advice when I needed one. I may not have come to Stanford without Shimon's help, and I am in his debt for that. The algorithms in this thesis were implemented using a code infrastructure called Frog or Phrog. Frog was initially created by Lise Getoor and myself, and I am convinced that had we known what we were getting ourselves into, we would not have done it. However, now that the beast walks among us, it is my hope that the various users of Frog actually bene ted from the system, despite all the pain that it caused. I would like to thank Eric Bauer, Xavier Boyen, Maricia Scott, Ben Taskar and Drago Anguelov for their help in writing Frog. Being a member of DAGS (Daphne's Approximate Group of Student) was a source vii of both pleasure and pride. Year after year, the research coming out of DAGS is of high quality and high impact, which is a testament to the extraordinary people that were and are members of DAGS, as well as to Daphne's vision and guidance. I would like to thank Drago Anguelov, Eric Bauer, Jenny Berglund, Xavier Boyen, Urszula Chajewska, Barbara Engelhardt, Raya Fratkina, Lise Getoor, Carlos Guestrin, Manfred Jaeger, Alex Kozlov, Brian Milch, Uri Nodelman, Dirk Ormoneit, Ron Parr, Avi Pfe er, Mehran Sahami, Maricia Scott, Eran Segal, Christian Shelton, Ben Taskar, and Simon Tong. I would like to thank some other friends and colleagues that I was fortunate enough to get to know along the years: Eyal Amir, Nikolaj Bj rner, Michael Colon, Arturo Crespo, Alon Efrat, Mattan Erez, Bernd Finkbeiner, Nir Friedman, Karl P eger, Henny Sipma, and Tomas Uribe. I would also like to thank my di erent sources of funding: ONR Young Investigator (PECASE) under grant number N00014-99-1-0464, ARO under the MURI program, \Integrated Approach to Intelligent Systems", grant number DAAH04-961-0341, ONR under the MURI program \Decision Making under Uncertainty", grant number N00014-00-1-0637, and NASA under grant number NAG2-1337. Finally, and above all, I would like to thank my parents, Tamar and Eliahu Lerner whose love and support were always something I could count on, and my brother Assaf who was always so proud of me | it is I who is proud to have you as a brother. This dissertation is dedicated to them. viii