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Learning High-Dimensional Markov Forest Distributions - Stochastic ...

Learning High-Dimensional Markov Forest Distributions - Stochastic ...

Learning High-Dimensional Markov Forest Distributions - Stochastic

Journal of Machine Learning Research 12 (2011) 1617-1653 Submitted 5/10; Revised 2/11; Published 5/11 Learning High-Dimensional Markov Forest Distributions: Analysis of Error Rates Vincent Y. F. Tan Department of Electrical and Computer Engineering University of Wisconsin-Madison Madison, WI 53706 Animashree Anandkumar Center for Pervasive Communications and Computing Electrical Engineering and Computer Science University of California, Irvine Irvine, CA 92697 Alan S. Willsky Stochastic Systems Group Laboratory for Information and Decision Systems Massachusetts Institute of Technology Cambridge, MA 02139 VTAN@WISC.EDU A.ANANDKUMAR@UCI.EDU WILLSKY@MIT.EDU Editor: Marina Meilă Abstract The problem of learning forest-structured discrete graphical models from i.i.d. samples is considered. An algorithm based on pruning of the Chow-Liu tree through adaptive thresholding is proposed. It is shown that this algorithm is both structurally consistent and risk consistent and the error probability of structure learning decays faster than any polynomial in the number of samples under fixed model size. For the high-dimensional scenario where the size of the model d and the number of edges k scale with the number of samples n, sufficient conditions on (n,d,k) are given for the algorithm to satisfy structural and risk consistencies. In addition, the extremal structures for learning are identified; we prove that the independent (resp., tree) model is the hardest (resp., easiest) to learn using the proposed algorithm in terms of error rates for structure learning. Keywords: graphical models, forest distributions, structural consistency, risk consistency, method of types 1. Introduction Graphical models (also known as Markov random fields) have a wide range of applications in diverse fields such as signal processing, coding theory and bioinformatics. See Lauritzen (1996), Wainwright and Jordan (2003) and references therein for examples. Inferring the structure and parameters of graphical models from samples is a starting point in all these applications. The structure of the model provides a quantitative interpretation of relationships amongst the given collection of random variables by specifying a set of conditional independence relationships. The parameters of the model quantify the strength of these interactions among the variables. c○2011 Vincent Tan, Animashree Anandkumar and Alan Willsky.

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